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BRIOT AND BOUQUETS 


ELEMENTS 


OF 


ANALYTICAL GEOMETRY 


OF TWO DIMENSIONS 


The Fourteenth Ldttton 


TRANSLATED AND EDITED 
1235'6 


JAMES HARRINGTON BOYD 


INSTRUCTOR IN MATHEMATICS IN THE UNIVERSITY OF CHICAGO 








\ & RAP y me 


| OF THE 
UNIVERSITY ) 


Tere 






d 


CHICAGO NEW YORK 
WERNER SCHOOL BOOK COMPANY 


) | eq | 


CopyYyRIGHt, 1896, 
By WERNER SCHOOL BOOK COMPANY. 


7646 6 





Typograpny BY J. 8. Cusuine & Co., Norwoop, Mass, 


PREFACE. 


Tus translation has been made with the hope that the 
high scientific character of Briot et Bouquet’s Lecons de 
Géométrie Analytique may contribute something toward the 
improvement of the standard of instruction in the elements 
of analytical geometry. 

The translator leaves for the second edition the addi- 
tion of notes which will bring some of the topics treated 
in the text down to the present scientific development of 
the subject. A note has been added with the object of 
furnishing the more elementary courses with simple exercises. 

I wish to thank Professor E. Hastings Moore, Professor 
Oscar Bolza, Professor Henry S. White, Dr. Harris Hancock, 
Dr. T. J. J. See, for valuable suggestions and assistance in 
making this translation. 

JAMES HARRINGTON BOYD. 


- UNIVERSITY OF CuHIcaGo, July 1, 1896. 


hii 


AUTHORIZED ENGLISH TRANSLATION. 


LIBRAIRIE CHARLES nee 


15, Rue Soufflot, a Paris. 


Adresse Télégraphique : 7 Soe G 
Paris. le b 


DELAORAVE PARIS 








Mrgerav— Le POON hig Ch ere By ue 
at pom’) Crrree D612y Cee a Arts « 


ee 





fe TiaRntor que vy fing de [omngi wn lope 


(Translation.) 
(DEAR SIR: 
As agreed to by both parties your translation will be regarded as the only authorized translation 
of this work (Briot et Bouquet Geometrie Analytique) in the English language. 
Accept, dear sir, my highest compliment, 
C. DELAGRAVE.) 









UNIVER ‘SITY 


——aw 


OF caLiFoRWS 


CONTENTS 


[Junior students will find all essential parts of the theory of Analytical 
Geometry of two dimensions included in those books and chapters not 


marked with asterisks. ] 


Boox I 
CHAPTER I 
CoNCERNING CO-ORDINATES. 


Rectilinear Co-ordinates. : , , : . : . 
Polar and Bi-polar Co-ordinates . 2 ; ‘ ° 
Representation of Curves in a Plane by Medations ; . P 


CHAPTER II 
EXAMPLES. 


The Circle. ‘ ‘ : : P : : : : 
The Ellipse 

The Hyperbola 

The Parabola 

Cissoid of Diocles 

The Straphoid 

Limagon of Pascal 

The Rose of Four Branches 

Tangents : 

The Conchoid ’ 

Pedals . : . 

ie ee 


CHAPTER III* 
~ ConceRNING HOMOGENEITY. 


‘Definition of Homogeneity . 

Remarks : 

Construction of Panwa ; : 

Irrational Formulas of the Second eae " 

Construction of the Roots of the Equation of the Scoend Degree 
1 


15 
16 
18 
19 
21 
23 
26 
31 
32 
36 
38 
39 


41 
47 


ae 


50 
51 


by 


CONTENTS. 


CHAPTER IV 
TRANSFORMATION OF CO-ORDINATES. 


Transformation of the Origin. F : 
- Change in the Direction of the Axes . 


General Transformation, and the Tyahekoemalion of octilinear oe 


ordinates into Polar Co-ordinates 
Distance between Two Points 
Harmonic Conjugate Points : : ; : ‘ : 
Classification of Plane Curves . i : ; ; : 


Boox II 


STRAIGHT LINE AND CIRCLE 


CHAPTER I 
STRAIGHT LINE. 


The Equation of the First Degree 

Meaning of Coefficients 

Equation of Straight Lines checuch a Gan Point 
Equation of a Straight Line through Two Given Points 
The Point of Intersection of Two Straight Lines 


The Condition that Three Straight Lines pass through a Point. 


The Angle between Two Lines ‘ 
The Perpendicular from a Point to a Line . ‘ 
The Equation of a Straight Line in Polar Co- ornate , 


‘CHAPTER II 
Tue CIRCLE. 


The General Equation of a Circle 

The Power of a Point with respect to a Circle 
Equation of a Tangent to a Circle 

The Radical Axis and Radical Center 

Pencil of Circles . ; ; : ; : 4 
Limiting Points . : . : : ‘ ‘ ‘ 
Orthogonal Circles 


Net of Circles and Equation ofa Circle in Polar Co- ordinates : 


67 
70 
72 
74 
75 
78 
81 
83 
88 


92 
94 
95 
100 
101 
102 
103 
106 


CONTENTS. 


CHAPTER III * 


GEOMETRICAL Loci. 


Geometrical Interpretation of the Equations. ; : . 
General Discussion and Remarks : ; : : : ; ° 
Examples of Geometrical Loci. 
Exercises 
Boox III 
CURVES OF THE SECOND DEGREE 
CHAPTER I 

CONSTRUCTION OF CURVES OF THE SECOND DEGREE. 
Genus Ellipse 
Genus Hyperbola . 
Genus Parabola 
Forms of the Polynomial of ihe Bacon Degree! in Two Variables 
Table of Classification of Curves of the Second Degree 
Discriminant of the Curve of the Second Degree 
Tangents to Curves of the Second Degree . 

CHAPTER II 
CENTER, DIAMETERS, AND AXES OF CURVES OF THE 
SeconD DEGREE. 

Definition of Center 
Diameters 
Conjugate Diameters 
Axes 


The Detendination of the Position of a Pont with ee ie a eon 


CHAPTER III 


REDUCTION OF THE EQUATION OF THE SECOND DEGREE. 


Ellipse and Hyperbola . 

Parabola ; 

Parameter of the Paola ; 

Invariant Expressions . 

Parameter of Loci of the Second pee 

Examples. : ‘ . . ° ‘ ; ° . 


PAGE 


109 
112 
113 
127 


182 
134 
138 
141 
143 
145 
150 


153 
157 
160 
162 
164 


168 
171 
173 
175 
176 
180 


4 CONTENTS. 


CHAPTER IV 
THE ELLIPSE. 
PAGE 
Equation in Normal Form (1) _ . : : : : ; ; . 188 
Axis and Vertices. : : : : : : ; . : . 184 
Polar Equation (8) _.. : ; : , 20 
The Ellipse is the Orthogonal Projecuon of a ce : : ; sae 
The Construction of the Ellipse by Points . ; . : : ee ty 
Equations of the Tangent and Normal ‘ : : : : a Ite 
Problems concerning the Ellipse . : : : : ; : Pee 0, 
Diameters of the Ellipse. ‘ : : : : . 193 
The Ellipse referred to Conjugate Dae oe : P ; ; . 195 
Theorems of Apollonius ; : , ‘ : : ; ; . 196 
Area of the Ellipse ; ; ‘ ; : : eh ee : . 200 
Equal Conjugate Diameters : : : ; ; ; : , 2ol 
Supplementary Chords : : : : , : : : . 202 
Exercises. : : 7 ; : ; : ; ; ‘ ear S Ys 
CHAPTER V 
Tue HyYPerBOoLa. 
Equation in Normal Form (1) . : : ‘ : ; : . 210 
Axes and Asymptotes . : ; ‘ : : : cee 4! 
Conjugate and Equilateral Eeperbeles : : : ‘ : whe 
Equations of the Tangent and Normal : ; : : : . 218 
Diameters. : A : . 215 
Hyperbola referred to Two Conus interes : ; : mer 4 
Theorems of Apollonius : : : . : : : ; . 218 
Construction of the Hyperbola . ; i : ; ; : . 220 
Supplementary Chords : : ; : oe re | 
The Hyperbola referred to its ery ; ; . : . 222 
The Area of a Hyperbolic ee ; : : : : : . 2238 
Exercises. ; : ; ; : , : : ‘ . 225 


CHAPTER VI 
CONCERNING THE PARABOLA. 


Normal Equation of the Parabola (1) and its Construction by Points 227 
Equations of Tangent and Normal . ; ; : ; : . 229 
Diameters. ; : : ° . : : . 2380 
The Area of a Parabolic Sone ; : . ; ; : . 252 
Exercises . ‘ ; : : ‘ ° . , ° ; . 233 


CONTENTS. 


CHAPTER VII 


Foci AND DIRECTRICES. 


Foci and Directrices of the Ellipse. : ; ‘ : . ‘ 

The Excentricity of the Ellipse . , ; : : ; : : 

Theorems concerning the Ellipse 

Convexity : 

Foci, Directrices and contig of the Hypsehols ; 

Theorems concerning the Hyperbola . 

The Focus of the Parabola . 

Theorems concerning the Parabola 

Transformation of the Equation of the Ellipse acd Batic io 
the Parabola . : . : : : j 

Theorems concerning the Ellipse or Hyperels : ; 

Trinomial Equation common to the Three Curves of the Scand 
Degree 

Polar Equations of ees of the Specad Degas 

Exercises 


CHAPTER VIII 
Tue Conic SECTIONS. 


Elliptical Section of a Right Circular Cylinder . 

The Equation of a Section of a Right Circular Gon is of the 
Second Degree 

Every Curve of the Second ieee a be placed on a Given Cone. 


CHAPTER IX * 
Ture DETERMINATION OF THE Conic SECTIONS. 


Five Conditions determine a Curve of the Second Degree in Two 
Variables ‘ ; 

Points and Imaginary Straight Lilies : 

Concerning the Intersection of Two Curves of the ena: Geme 

Equation of a Curve of the Second Degree which passes through 
Five Given Points . . : 

Abridged Forms of the manennn of the ore eee Multiple 
Conditions : : : : : 

General Method for dotorninine ine Foet of a Corie 

Investigation of Two Secants common to Two Curves of the Becca 
Degree 

The Number of Normals aa a Given Point me an Ellipse 

Exercises. : ° ‘ . . ; ° e ° ° 


279 


281 
283 


286 
287 
289 


296 


802 
304 


307 
312 
314 


6 CONTENTS. 


CHAPTER X* 


THEORY OF POLES AND POLARS. 


Harmonic Proportion . 

Polar and Pole 

Conjugate Straight Lines 

Autopolar Triangle — Conjugate Tr nae 
Reciprocal Polar Figures 

Degree or Order and Class . 

Envelope Curves . 

Examples 

Tangential Conair 

Tangential Equation of a Conic . 


CHAPTER XI* 
GENERAL PROPERTIES OF CONIC te 


Pascal’s Theorem 

Brianchon’s Theorem . 

Homographic Systems . ; 

Homographic Systems of Points . 

Homographic Systems of Straight Lines 
Involution , : ; 

Homographic Pencils ; 
Characteristics and Seay Co ace 
Points at Infinity — Straight Lines at Infinity . 
Formulas of Transformation : : 
Applications 

Homogeneous Co- ena of a Siright eee 
Trilinear Co-ordinates . 


. 
ft 
ec 


Equation of the Straight Line at Infinity in T ciicear Co- ora : 


The Equation of a Line passing through Two Points 
Tangents in Trilingar Co- ordinates 
Conjugate Conics . 


Examples 
Exercises 
CHAPTER XII 
Secants Common to Two Conics. 
Lemmas and Equation ind (1) . . ‘ : : 


Osculating Conics (3°) 
Sub-osculatory Conics . ; : ; : ‘ ; 


PAGE 
318 
320 
324 
326 
326 
328 
380 
537 
341 
345 


395 
401 
402 


CONTENTS. ff 


PAGE 
Examples. ‘ ‘ : ; 7 ‘ , : ; : . 403 
Exercises. : : : ‘ ‘ ‘ . 405 
Relations scmnesting the Roots in : : : ‘ : : . 406 
Theorems I., II., II. . q : : : . 409 
Conics Harmouloally Circumseribed and Toseribed ; : ee: 27” 
Examples. : ; : ; : : ; ; : F . 414 
Exercises. 415 
The Application of fie Bacperes of ifomogencous Polynomials io 
the Theory of Curves of the Second Degree : ‘ ; 9 0 
The Discriminant A. ; 417 
The Polynomial of the Second Degree in ce Form ot the Bam of 
Squares . : ; , : : : 4 ; ; . 418 
Geometrical Thteraretations ; : A A : : ; . 423 
Book iV 
THE GENERAL THEORY OF CURVES 
_ CHAPTER I 
Tur ConsTRUCTION OF CURVES IN RECTILINEAR CO-ORDINATES. 
Introductory Remarks. ; : : : : : ; . 428 
Examples I., II., IIL., IV., V., VIL ; : ; , : : . 430 
The Pntvoduotion of an eee Variable : ; : ; . 437 
Tangent and Orthogonal Curves . : ; : : ‘ ‘ . 440 
CHAPTER II 
ConvEXITY AND CONCAVITY. 
Definitions .. : ; : ; ; ; : . 443 
Examples I., IL., Ill, IV. y. : : : ; ; ; : . 445 
Algebraic Curves . : 449 
Singular Points, Point ve ét (447), Point satLeAt Simpl Point, 
Double Point, Cusp, Point of pth Order. ; : . 461 
Hessian, Non-singular Point. : ‘ : : ‘ . 455 
Point of Inflection : : : : : ; . 456 
Class of a Curve and Pliicker’ S Wgnaticns : ; : : ; . 457 
Curves in Trilinear Co-ordinates A ; : . ; Pp . 458 
CHAPTER III 
ASYMPTOTES. 
Definition . : : ; ; . 461 


Asymptotes which are eel S the . -axis ete : . 462 
Asymptotes which are not parallel to the y-axis ; F ; . 487 


8 CONTENTS. 


: PAGE 
Examples. ; 469 

Reduction of the Sindy, of ans Boceke to aia of Finite 
Branches ; 7 : ‘ : : y ; ‘ . 475 
Parabolic Branches. : : : : : : ; : . 476 
Asymptotic Curves. . : : : : ; : : . 479 

CHAPTER IV 
CoNSTRUCTION OF CURVES IN POLAR CO-ORDINATES. 

Spiral of Archimedes . : : : : ee ee : : . 481 
Tangent : : 484 

Equation of oe Bub- ae Sub- Oe) and ee of 
Normal . ; : : : : : . : : A . 486 
ExamplesI.-X. . : : 7 A : : ; : : . 487 
Convexity and Concavity . : : : ; ; : : . 491 
Asymptotes . ; : : ; ; : : : : : . 492 
Exercises. ; ° : ; ; : ; : : = oe OVA 

CHAPTER V 
CoNCERNING SIMILITUDE. 

Definitions . ; fae ; ; : : : . 504 
The Equation of Homiothetie Care: : : : : ; ; . 506 
General Equation of a Species of Curves . ; : ‘ : - 610 
Condition of Similitude of Two Figures. ; : ; ; . 514 


CHAPTER VI 


GRAPHIC SOLUTION OF EQUATIONS. 


Solution of Equations of the Third and Fourth Degrees. ; . 518 
Examples E., IL, IL, IV. . P : ; ‘ ; “i . 620 


CHAPTER VII 


Notions CONCERNING UnicursaL CURVES. 


A Curve of the mth Order with a Multiple Point of the Order m — 1 


is Unicursal . ; ‘ ; ; , : : . 523 
Unicursal Curves of the Third Ondes ; : : ; ‘ . 524 
Unicursal Curves of the Third Order with One Case: ; , . 625 
Curves of the Third Order with a Double Point ; : : << BBs 
Exercises. : ee oe ; A : : ; ; : . 532 
Norte, with Examptes I.-IX. . : ‘ ; : ; : . 537 


EXAMINATION QUESTIONS . ‘ . ‘ p ° : ‘ . 557 


LIBRARY 
Or TEE 


UNIVERSITY 










OF CALIFORN Nw 


ANALYTICAL GEOMETRY 


——209300— 


ANALYTICAL GEomeEtry has for its object the study of fig- 
ures through the methods of algebraic calculation or analysis. 

The representation of figures by’ algebraic symbols is due 
to Descartes, who established a general method for the resolu- 
tion of geometrical questions. 

In this treatise, plane figures, or those of two dimensions, 
are considered. 





PLANE GEOMETRY 


Boox I 


CHAPTER I 


CONCERNING CO-ORDINATES. 


The position of a point in a plane is determined by means 
of two magnitudes, which are called the co-ordinates of that 
point. 

There can be an infinity of systems of co-ordinates. An 
exposition of those systems only is given which are most 
simple and most used. 


RECTILINEAR CO-ORDINATES. | 


1. Let there be two non-parallel straight lines or fixed axes 
X'X and Y'Y traced in a plane (Fig. 1); the position of any 
9 


10 PLANE GEOMETRY. BOOK I. 


point M of the plane will be determined by the intersection of 
the two lines G'G, H'H parallel to the axes. The position of the 
parallel H'H is defined by the segment OP, which it intercepts 
on X'X. It is necessary to indicate the 

i fis direction in which the length OP is 

3 jr ¢@ measured. For this purpose it will be 














a, convenient to give the sign + to the 
| alas distance OP, if it is measured on OX, 

x O Ae Ds . a ae 
/ for example; the sign —, if 1t 1s meas- 
7 je ured on OX’. In like manner, the posi- 
ee tion of the parallel G'G is defined by 


the length OQ affected with the + sign 
or the — sign, according as it is measured on OY or OY". 

The two lengths OP and OQ (each affected with the proper 
sign), which determine thus the position of the two parallels, 
and consequently the point M, are the rectilinear co-ordinates 
of M. They are usually represented by the letters « and y. 
Further, the co-ordinate designated by w is given the name 
abscissa; the other, y, that of ordinate. The two fixed right 
lines X'X and Y’Y are called the axes of co-ordinates ; the 
first is the axis of the a’s, and the second the axis of the 
y’s. The point O from which we measure the co-ordinates on 
each axis, in the one direction or in the other, is called the 
origin of co-ordinates. 

If all possible values, positive or negative, be assigned to # 
and to y,—in other terms, if # and y be made to. vary from 
— we to +, —all points of the plane are obtained; otherwise, 
each pair of values gives a point, and one only. 

The two co-ordinates of the point M are the projections of 
the line OM, taken in the direction OM, on the axes OX and 
OY, the projection on each axis being taken parallel to the 
other. The projection on the axis of is the length OP, 
identical with the co-ordinate 2, affected with the + sign or 
— sign, according as it 1s measured in the direction OX or in 
the opposite direction OX"; the projection on the axis of ¥ 
is the length OQ identical with the co-ordinate y, affected 
with the + sign or — sign, according as it is measured in 
the direction OY or in the opposite direction OF" 


CHAP. I. CONCERNING CO-ORDINATES. HH 


RECTILINEAR RECTANGULAR CO-ORDINATES. 


2. Usually the fixed axes are drawn per- Y 
pendicularly to each other; in this case, the 
two co-ordinates of the point M (Fig. 2) are wi 
the distances of this point from the two A 
axes: they are also the orthogonal projec- 
tions of the line OM on the two axes. 














i Fig. 2. 
POLAR CO-ORDINATES. 


3. Let O be a fixed point called the pole, OX a fixed axis 
(Fig. 3). We can determine the position of a point M by the 





length p, the radius vector OM, and by the a 
angle w, which the radius vector makes with p 
the axis. . oe 

The position of the point Mis determined ? x 


‘by the intersection of a circle of radius p, 
having the pole for center, and the half-line OL drawn from 
the pole and making the angle w with the axis OX (Fig. 4); but 
it is necessary to define the direction in 
which we reckon the angle w, namely coun- 
ter clockwise from the axis OX. All the 
points of the plane are obtained if p vary 
from 0 to +0, and w from 0 to 27. In 
fact if, w remaining constant, one makes p 
vary from 0 to +o, one has all the points 
of the half-line OL; if, therefore, » vary from 0 to 27, the 
half-line OZ moves from the position OX and describes the 
entire plane. 








Fig. 4.) 


BI-POLAR CO-ORDINATES. 


4, The position of a point may also be defined by the dis- 
tances wu and v from two fixed points F and F" (Fig. 5). The 
position of the point M is then determined by the intersection 
of the two circles described about the points # and F” as cen- 
ters with the radii uw and v. However, this system does not 
offer the same theoretical perfection as the two preceding; for 


12 PLANE GEOMETRY. BOOK I. 


every couple of values of wu and v is not admissible; it is neces- 
sary that the distance between the poles be less than their 
sum and greater than their difference. 


2 When this condition is fulfilled, the 
L(™\ two circumferences intersect in two 
\ points, and a troublesome ambiguity 
x arises. 
ne The position of the point M may 
a still be determined by aid of the an- 


gles MFF", MF'F; we designate these angles, reckoned in a 
definite direction, by a@ and 8; each of them varying from 0 
to 24: to every couple of values of « and B corresponds one 
point of the plane, and one only. 


5. The number of systems of co-ordinates is infinite. In 
general, the position of a point in a plane is determined by 
the intersection of two lines traced in 
this plane. Let A’, A", A'",--- (Fig. 
6) be a first system of lines of the 
same kind, corresponding to the sev- 
eral values wu’, u"', wl", «++ of the variable 
; u; B', B", Bl,..- a second system of 
: lines of the same kind, correspond- 

ing to the several values v', uo", ull, aes 
of the variable v; any arbitrary point of the plane is defined by 
the two lines which meet in this point, and the particular 
values which it is necessary to give to the vatiables w and 
v, in order to determine these two lines, are called the co-ordi- 
nates of the point. The totality of these two series of lines 
constitutes a system of co-ordinates. 

In the first system which we have studied, each series of lines 
is composed of right parallel lines; hence the co-ordinates were 
given the name rectilinear co-ordinates. 

In the polar system, the first series of lines is composed of 
half-lines emanating from the pole O, and positioned by the 
variable angle , which they make with the axis OX (Fig. 4); 
the second system of concentric circles described about the 
point O as center with the variable radius p. 





 “Fig.6° | 


CHAP. I. CONCERNING CO-ORDINATES. 18 


In the first bi-polar system each series is composed of con- 
centric circles (Fig. 5). In the second, each series is composed 
of half-lines emanating from one of the points F' or F". 


REPRESENTATION OF LINES IN A PLANE BY EQUATIONS. 


6. Let AB be any line whatever, straight or curved, in the 
plane (Fig. 7); draw in the plane two straight lines OX and 
OY, and designate by x and y the two co- 
ordinates OP and PM of any point M of 
the line; when the point 7 is moved along 
the line, the two co-ordinates vary simulta- 
neously; if an arbitrary value be assigned 
to OP, the magnitude of the corresponding , 
ordinate MP is perfectly determined, and Fig. 7. 
the variation of the abscissa controls that of the ordinate. That 
is, the ordinate MP is a function of the abscissa OP; the char- 

, acter of this function depends on that of the line. If the line is 
defined geometrically, an equation between x and y, serving to 
define the function y, can be deduced from the geometric defini- 
tion of the line. The equation which is found in this manner 
is called the equation of the line. 











7. Conversely, let there be given an equation 
ia, y= 0, 

between the variables x and y; each pair of real values of & 
and y, satisfying this equation, determine a point of the plane. 
Let a and y be a pair of real values of w and y satisfying 
the equation; if x begin with the value 2%, and vary in a con- 
tinuous manner, one of the values of y, beginning with y, will 
also vary in a continuous manner, and will in general be real 
as long as a is restricted to varying between certain limits: 
the point, of which the co-ordinates are w and y, will describe 
in the plane a continuous line. Thus, the totality of the real 
solutions of an equation in two variables is, in general, repre- 
sented by a line in a plane. 


8. What has been said concerning rectilinear co-ordinates is 
applicable to every other system of co-ordinates. In the polar 


14 PLANE GEOMETRY. | BOOK I. 


system, where the point M is on the given line, the radius 
vector p varies with the angle w; it is a function of o, and the 
line is represented by some equation between p and wo. 


9. The representation of figures by equations is the object 
of Analytical Geometry, and in it the results of algebraic cal- 
culation are applied to their study. In Analytical Geometry 
the student is occupied with three fundamental questions: 
given a figure defined geometrically, determine its equation; 
conversely, given an equation, determine the figure which 
corresponds to this equation; finally, study the relations which 
exist between the geometric properties of the figures and the 
analytic properties of the equations. 

The examples which are given in the following chapter will 
show how lines may be represented by equations. 


~ 


CHAP. II. EXAMPLES. 15 


CHAPTER II 
EXAMPLES. 


In general, the geometric definition of a curve determining 
each of the points corresponds to a certain system of co-ordi- 
nates; if the particular system implied by the definition be 
chosen, the equation of the curve is the immediate algebraic 
translation of its geometric definition. 


CIRCLE. 


10. The circle is the locus of all points equally distant from a 
jived point called the center. It is described by means of a com- 
pass; one foot being placed at the 
center, the other will trace the circum- 
ference. | 

If the center O be taken as pole, and 





and r represent the length of the radius, 
the equation of the circumference in 
polar co-ordinates is Fig. 8. 


(1) PS * 
since the length of the radius vector is constant and equal to r 
whatever value the angle w may have. 

Let us now seek the equation of the circle in rectilinear 
co-ordinates. If two rectangular axes OX and OY passing 
through the center be taken, the right triangle OMP gives 
immediately the relation 


2) e+ yar, 
which exists between the two co-ordinates x and y of any point 


M of the circumference. This is the equation of the circum- 
ference in this system of co-ordinates. 





any right line OX as polar axis (Fig. 8), <3 . 2 


16 PLANE GEOMETRY. BOOK I. 


ELLIPSE. 


1l. The ellipse is a curve such that the sum of the distances of 
each of its points from two fixed points is constant. The two 
fixed points are the foci of the ellipse. 

Let 2a represent the sum of the distances of any point of 
the ellipse to the foci, and 2¢ the distance FF" between them. 
The points of the ellipse can be constructed by describing a 
circle with arbitrary radius u about one of the foci as center, 
and a second circle about the other focus as center with radius 
v equal to 2a—u. The points of intersection M and M' of 
the two circles belong to the ellipse. In order that the two 
circles intersect, it is necessary that the longest radius be no 
longer than a +’c, and the shortest no shorter than a — c. 

The points Mand M' being symmetrical with respect to the 
line FF’, this line is an axis of curve. The right line BB’, 
perpendicular to FF" at its mid-point O, is a second axis. 

The points in which the axes cut the curve are called swm- 
mits. The summits A and A’ are obtained by taking the 
distances F'A, F'A' equal to a—c. The summits B and B, 
situated on the second axis, are deter- 
mined by describing a circle with radius a 
about one of the foci as center of a circle. 
The distance OA is equal to a, and the 
distance OB, which is designated by 8, is 
equal to Va?—c.. Instead of defining 
the ellipse by the lengths 2'a and 2c, as 
in the preceding, it can be defined by 
means of the lengths 2a and 2b; hence,c=Va?— 6°. The 
point O, the mid-point of FF", is the center of the curve. 








12. We now derive the equation of the ellipse. The system 
of co-ordinates used is the first bi-polar system (§ 4); if the 
position of each of the points of the plane is determined by 
its distances from two fixed points F and F’, the ellipse will 
have for its equation, 


(1) Uutv=2a. 


CHAP. II. EXAMPLES. 17 


In the second bi-polar system the equation has also a very 
simple form; if « and 8 represent the two co-ordinate angles 
MFF', MF'F, and 2p the perimeter of the triangle MFF"', then 


ng —~|(P— 26) (p—%), nf = (p—2c)(p—v), 
» =\' p(p —) Ws . p(p—u) ” 














2 

2c _a—c 
2 tan tan ® —P — 
(2) an 5 aaa Higa ser 


13. Finally, the equation in rectilinear co-ordinates is con-. 
sidered. Take the two axes of the 

















¥ 
curve as axes of co-ordinates (Fig. ‘i 
10); the lengths PF and PF" being 
equal to c— a and c+4a, the right- a . 
angled triangles FMP, F'MP, give ry o| PF 5 
w=Vy + (c— 2)’, Nae ee 
—  vevyt (c+ 2), | Fig. 10. 


By substituting the values of uw and v in equation (1), we 
obtain the equation 


6) yt(c—aPr+vVyt (c+ 2)? = 2a. 


Transposing the first radical to the second member and 
squaring, gives 


yYt+(ctorv=4eePt+yt+(e— ae Vr anarmey 


or, simplifying, 














avy + (¢ —@)? = a — cw. 
Squaring and transposing lead to the equation, 
(4) . a’y? + (a? — &) a? = a? (a? — c’). 


However, equation (4) is not equivalent to equation (3); it 
is equivalent to the four equations 


ris 


u+tv=2a, u—v=2a, —u+v=2a, —u—v=2a, 


which are obtained from equation (3) by changing the sign of 


the radicals. The equation — uv — v= 2a has no real solution. 
B 







\ 1B RAB 
OF THE 


UNIVERSIT 





CF CALIFORWIK 


18 PLANE GEOMETRY. : BOOK I. 


The equation vu —v=2a, and —w+v= 2a, do not have real 
solutions if one suppose 2a>2c; because the quantities wu 
and v represent the distances of the points / and F" from 
a point in the plane whose co-ordinates are w and y, and the 
difference of these distances cannot be equal to the length 2a, 
greater than the distance 2c or FF’. Thus, when real solu- 
tions only are considered, the equation (4) can be regarded 
as equivalent to equation (3). The constant sum 2a being 
greater than the distance between the foci 2c, one can put 
a? — c? =b?, and the equation of the ellipse reduces to the form 


a®y” + bx? = a’b*, or 


a 2 
(5) pt B =1. 
HYPERBOLA. 


14. The hyperbola is a curve such that the difference of the 
distances of each of its points from two fixed points is constant. 
The two fixed points F and F" are the foci of the hyperbola. 

The hyperbola, like the ellipse, has two axes. of symmetry, 
the right line, FF’ (Fig. 11), and the perpendicular, BB’, to 
B this line at its mid-point O. 
It is composed of two distinct 
branches. The points of a 
branch of a hyperbola can be 
constructed by describing a 
circle with an arbitrary radius 
wu about F as a center, and a 
second circle with a radius v 
B' (a equal to 2a + u about F' as a 

Fig. U1. center. In order that these 
circles intersect, it will be necessary that u be greater than 
e—a. In a similar manner a second branch may be found. 
The point O, the middle of FF’, is the center of the curve. 

The first axis intersects the curve in two points only, namely 
A and A’, which are its vertices and are determined by taking 
0A =OA!' = a; this axis is for this reason called the transverse 


axis. 


N, 
/ - 

“7 

/ 

U 








ah 


f 
! 
‘ 
‘ 
! 
1 
' 
1 
! 
Al 
Di 
' 
4 
4 
\ 
\ 
s 
\ 





CHAP. II. ‘ EXERCISES. — 19 


In the first bi-polar system, if u and v represent the distances 
of any point of the curve from the foci ¥ and Ff", the two 
branches of the curve have respectively for their equations 
(1) v—u=+2a4a. 


In the second bi-polar system the equations of the two 
branches are 











(2) 3 c+a sae c—a 
? = ) = . 
tan aa tan 2 ang 


15. If the two axes of the curve are taken as axes of 
co-ordinates, the equation of the hyperbola in rectangular 
co-ordinates will be 


y+ era —VvVe+ C—4#)'= + 2a. 


By repeating the transformation of (§ 13), one obtains the 
integral equation a?y? + (a? — ¢) # = a? (a@ — c’), which we have 
already obtained for the ellipse. 

This equation, as has been remarked, is equivalent to four 
distinct equations v—-uw=+2a,u+v=+2a; but in the 
given case 2a being smaller than 2c the last two equations 
have no real solution. Placing c?—a?= 0’ the equation 
becomes 








It is well to observe that, in the rectilinear system, the two 
branches of the hyperbola are embraced in the same equation 
(3), while in the first bi-polar system, one of the branches is 
represented by the equation v —u = 2a, the other by u—v= 
2a. It is also necessary to have two distinct equations in the 
second bi-polar system. 


PARABOLA. 


16. The parabola is a curve every point of which is equally 
distant from a fixed point called the focus and a fixed line called 
the directrix. : 


20 


PLANE GEOMETRY. BOOK I. 


The perpendicular drawn through the focus to the directrix 

















Fig. 12. 








is an axis of symmetry of the curve. The 
point A, middle of DF, is the vertex of 
the parabola. The curve lies wholly to 
the right of a line drawn through 4A, 
parallel to the directrix. Any point of 
the curve can be found by drawing a 
line MM' parallel to the directrix, at the 
distance DP greater than AD and describ- 
ing a circle with a radius equal to this 
distance DP about the focus as center 
(Fig. 12). 


17. The definition of the parabola suggests a system of co- 
ordinates, which has not yet been considered. Any point, M, 


be 


E 








fe nn ae ae ae a ee 








Fig. 13. 


of-the plane can be determined by the dis- 
tances MF and ME from the fixed point 
F and fixed line DD! (Fig. 13). The posi- 
tion of the point M will be determined by 
the intersection of a circle described about 
Fas a center and a right line parallel to 
DD'. Tf we call wand v the two co-ordi- 
nates of the point M, the parabola will 
have for its equation, in this system, 


U= Uv. 


18. Let A, the vertex of the parabola, be,taken for the 
origin of rectangular co-ordinates, the axis of the parabola for 
the v-axis and the perpendicular AY for the y-axis. Repre- 
sent the distance FD of the focus from the directrix by p: 


then is 





2 
(1) v= AP + AD=2 +4, w= vt (2-3), 


and the equation of the parabola becomes 


or (2) 





2 
V7+(#~4) =a+%; 


y? = 2 pe. 


CHAP. II. EXERCISES. 21 


19. Before proceeding further the definition of a tangent to 
any eurve whatever will be given. In elementary geometry, 
a line is said to be tangent to a circle when it has but one point 
in common with the circumference, but this definition cannot 
be generalized and_it will be convenient to define a tangent in 
another manner. Let M be a 
‘given point on a curve (Fig. 
14); through this point and a 
neighboring point M' draw an 
indefinite right line; in the figure which we study, the direc- 
tion MM’ has, in general, a limiting position MT, as the point 
M' approaches the point Mas alimit. The right line MT is 
called a tangent to the curve at the point M. The perpendicular 
to the tangent at this point M is called the normal to the curve. 

From this definition, it follows that the tangent to the circle 
at the point M is perpendicular to the radius OM at its ex- 
tremity (Fig. 15); because, in the isosceles 








triangle MOM", the angle OMM' is equal Se T 
to aright angle less half the angle MOM". a 
When the’ point M’' approaches contin- 

uously toward the point M, the angle at 0 


the center approaches zero, and the angle 
OMM' becomes right-angled. The normal 
to the circle in M is the radius MO. Fig. 15. 


CISSOID OF DIOCLES. - 


20. If one be given a circle, a diameter AB, a tangent BC 
at the extremity of this diameter (Fig. 16), and if a secant 
AE be made to revolve about the point A, on which a length 
AM be then equal to the distance DE comprised between the 
circle and the fixed tangent, the locus of Mis a curve which is 
called the cissoid. 

If the movable secant start from the position AB and 
revolves about the point A, from AX toward the perpendicular 
AY, the length DE, and consequently AM, increase indeti- 
nitely, and the point M will describe an infinite branch MM" of 
the curve. By revolving the movable secant from the other side 


y as PLANE GEOMETRY. BOOK Tf. 


of AB, a second branch of the curve equal to the first is 
obtained. The line AB is an axis of 
\ the curve, since the two branches are 
symmetrical to this right line. 
The tangents to the two branches 
ee, at the point A coincide with the axis. 
Because, if the secant revolves about 
D the point A in such a manner that 
\ the chord AM or DE becomes zero, 
\\ it tends toward the limiting position 
5B  <. AB; therefore AB is the tangent at 
me. A. The point A is called a cusp ‘or 
turning point. It is also apparent 
that the two branches of the curve 
continually approach the line CC". 
e In fact, consider the secant in the 
position AH'; if from the line AE’ 
sie the equal lengths AM’ and D'E' be 
subtracted alternatively, then will M'H'= AD'. The chord 
AD! diminishes continually and approaches zero; it is equal 
in length to M'E', therefore for a greater reason does the per- 
-pendicular M'H approach zero. The right line CC', which 
the curve continually approaches, is called the asymptote. 
The cissoid was conceived by the Greek geometer, Diocles, 
to solve the problem, to construct two mean proportionals 
between two given lines. 


g 21. Let us seek the equation of the 
if cissoid in polar co-ordinates; take the 

point A as pole and the right line AB 
for the polar axis. Call a the diameter 
of the given circle, p and w the co-ordi- 
nates of any point M‘of the curve (Fig. 
17). In the right-angled triangles ABE, 
Fig. 17. ABD, one has 


AE=—“., AD=acoso; 
COS w 


whence, p= DE = AE — AD= 


E 














eG 

















a asin? w 
— £608 o'= ———_— 
COS w COS w 


CHAP. IL. _ -EXAMPLES. ee 


Hence the cissoid has for its polar equation, 


(1) 


Let us now derive the equation in rectangular co-ordinates ; 
take the point A for the origin, the right line AB for the 
w-axis, and a perpendicular for the y-axis. From the triangle 
MAP, one gets 


a= pcose, y=psino, p=w+y’; 


_ asin’? w 
COS w 


if, in equation (1) cosw be replaced by * sinw by Y. it 
p 


p 
becomes px = ay’, then p? by 2 +7’, the equation of the 
cissoid in rectangular co-ordinates will be 


" (2) y(a— a2) —a = 0. 


22. Having already derived the equation of the cissoid in 
rectangular co-ordinates from its geometric definition, it is 
proposed to construct the curve from its equation. Solving 
the equation (2) with respect to y, one has 


ce 


Go 





The ordinate is real for all values of the abscissa comprised 
between «= 6 and «=a and for those values only ; therefore 
the curve lies wholly between the y-axis and the parallel CC’ 
erected at the distance a (Fig. 16). As x increases from 0 to 
a, the numerical value of y increases from 0 to o, which 
determines a branch of the curve beginning at the origin A 
and ascending indefinitely. At the same time the distance 
MH =a-—zx of a point on the curve to the line BC approaches 
zero, which shows that the line BC is an asymptote of the 
curve. Since to each value of # there corresponds two equal 
values of y opposite in sign, the curve is composed of two 
branches symmetrical with respect to the axis AX. 


STROPHOID. 


23. A right angle YOX (Fig. 18) and a fixed point A on 
one of the legs, being given in a plane, draw from the fixed 


24 PLANE GEOMETRY. BOOK Tf. 


point A any line AD, which cuts the side OY in D, and begin- 
ning at D, lay off to the one side and to the other on this line, 
H aan mag Fe the lengths DM and DN equal to 
OD; the locus of the points M and 

N ie the strophoid. 
Ny When the movable line occupies 
D the position AO, the two points M 
M and N fall together in O. If the 
= ‘ aa line moves so that the point D 
A rz ascends continually on OY, OD in- 

N 





creases and the point NV describes 
the infinite branch OW of the curve. 
The point M approaches continually 
the point A, because the points M 
H’ x K’ and N are obtained by describing 

Fig. '18. a circle with radius DO about D 
as a center; as the point D recedes continually from O, the 
angle OAM approaches a right angle and the point M coin- 
cides with A. . The curve has evidently another branch sym- 
metrical to the first with respect to the axis OX. 

The point O, in which the two branches of the curve cross, 
is a double point. The tangents to the two branches of the 
curve at this point coincide with the bisectors of the angles 
YOX and YOX'. Because the angle ODE, exterior to the 
isosceles triangle DOM, is equal to the sum of the two oppo- 
site interior angles and, consequently, to two times the angle 
DOM;; similarly the angle ODA is equal to two times the angle 
DON. As the line AD approaches OA, the obtuse angle ODE 
decreases and tends toward a right angle; therefore the half 











angle YOM decreases and tends toward 7 The acute angle 


ODA increases and tends toward a right angle; hence the half 


angle YON increases and approaches ; as its limit. Whence 


it follows that OM and ON are perpendicular to each other 
in their limiting positions as tangents at O. It is to be noticed 
further that the are OMA is below, while the are AON is 
above its tangent. 


CHAP. II. EXAMPLES. 95 


The tangent at A is perpendicular to the axis OX, because 
as the point D is continually elevated, the cnt AM becomes ~ 
finally perpendicular to OX. 

On AO produced take OG = OA, and at the point G erect 
the perpendicular HH'. This line is an asymptote to each of 
the infinite branches of the curve, for the distance NE, equal 
to AM, approaches zero. 


24. To derive the equation of the curve in polar co-ordi- 
nates, take the point O as pole and the line OA as polar axis; 
the polar co-ordinates of the point Mare p= OM, o= MOA; 
in the isosceles triangle DOM, each of the angles DOM, DMO 


is equal to a w, and the angle ODM to 2; the angle OAM, 
complement of the preceding, equals ‘7 2. 


If a represent the length OA, it follows from the triangle 








OMA that 
: sin i ae ») 
a : 
sin @ i ) 
whence 
acos 2w 
(1) C ebaw 


The co-ordinates of the point WN satisfy the same equation. 
To derive the equation of the curve in rectangular co-ordi- 
nates, take for axes the two lines OX and OY. If, in the pre- 
ceding equation, put under the form p cos w =a (cos’ w — sin’ w), 
a 
cosw and sinw be replaced by their values a 2 one gets 
ap? = a (a? — y*); putting in the place of p’ its value 2 + y’, the 
following equation of the third degree is obtained : 
(2) a(x? + y’) — a(x — y’) = 90. 


25. The curve can now be constructed by means of its equa- 
tion in rectangular co-ordinates. Equation (2), solved with 


respect to y, becomes 
ie lau —2 
= a+2 





26 PLANE GEOMETRY. BOOK I. 


In order that the ordinate y be real, it is necessary that the 
quantity under the radical be positive. If a be given positive 
values, the denominator being positive, the numerator will also 
be positive so long as 2 is less than a. If & be given negative 
values, the numerator being positive, the denominator will 
be positive so long as the absolute value of a is less than a. 
Thus the abscissa 2 can vary from —a to +a. If, therefore, 
we begin at the origin and lay off on the a-axis, to the right 
and to the left, the distances OA and OG equal to a, and at 
the points G and A erect HH', KK' parallel to the y-axis, the 
curve will be wholly comprised between these two parallels. 
The form of the curve will vary in accordance with the varia- 
tion of the function 


ee ja—@ ; 
Coc a-e 


As avaries from 0 to a, the ordinate y takes finite values. 
It is zero for e=0 and also for =a. This furnishes the 
branch OMA of the curve, beginning at the point O and end- 
ing in the point A. As @ varies from 0 to — a, the ordinate y 
is negative and varies from 0 to —#. This furnishes the 
branch ON', which begins at the origin and descends con- 
tinually, approaching indefinitely the line HH', which is an 
asymptote. This branch ON’ is a continuation of the branch 
AMO. 

By changing the sign of the radical, the branch AM'ON, 
symmetrical to the first with respect to the w-axigy 1s obtained. 


LIMACON OF PASCAL. 


26. Through a point A on a circle, draw any secant AD, on 
which beginning at D, where it cuts the circle again, lay off a 
constant length DM or DN; the locus of the points M and 
(Fig. 19) is a curve called the Limagon of Pascal. 

The entire curve will be traced by supposing the radius 
vector to coincide with the diameter AB of the circle and then 
to revolve through an angle z in either direction. The whole 
curve will also be traced by giving the radius vector a com- 


CHAP. IL. EXAMPLES. 3 7 


plete revolution, and laying off a constant length in the direc- 
tion of radius vector, beginning with the point in which this 
radius vector or its pro- 
- longation intersects the oe! u 
circle. The curve takes ; 
three different forms ac- 
cording as the constant 
length a is greater, ,equal s 
- to,or less than the diam- * 4/74 B Gs 
eter b of the circle. 

1° The first case con- 
sidered is when the length 
a is greater /than O 4E 7 
the radius vector coin- 
cides with AB, it will be 
necessary to start from the point B. Construct on AB a length 
BG equal to a, which determines the point G of the locus (Fig. 
19). If the radius vector revolves from the point 4 and takes 
the direction AD, the point M is determined. When the 
radius vector has revolved through a right angle, the point D 
coincides with A, and the point M with M’. Continuing the 
revolution of the radius vector to the position AD! and pro- 
longing it, it intersects the circle in D,; it is necessary to start 
from this point D,, and lay off in the same direction AD’, a 
length D,M, cqual to a. When the radius vector having 
revolved through the two right angles occupies the position 
AX', the point D, coincides with B and the point with H; 
thus the arc, M'M,H, a continuation of GMM’, is constructed, 
and is, moreover, exterior to the circle. The radius vector 
revolving beyond AX' through two more right angles returns 
to its initial position AX; the moving point describes the arc 
HN'G, symmetrical to the are GMH, with respect to the line 
XX". In this manner, by a continuous movement, the point 
describes the entire curve. 

2° Suppose that the length a be equal to 6. When the 
radius vector, starting from the initial position AX, moves 
through two right angles, the point M describes the arc 
GMM'A (Fig. 20), which ends in the point A. The tangent 








Fig. 19. 


28 PLANE GEOMETRY. BOOK I. 


at Ais the right line AX’, limiting position of the secant 4M,. 
The point A is called a cusp. 

3° As the last case to be considered, let a be less than b. 
When the radius vector, starting from the initial position, has 














Fig. 20. 


revolved through one right angle, the point M describes the 
arc GMM' (Fig. 21). If now the radius vector takes the di- 
rection AD’, the point D moves to D,, the point M falls in M,. 
But when the radius vector assumes a direction AD", such 
that the chord D,A is equal to a, the point M, falls then in A, 








- Fig. 21. 


and the curve will be tangent to the right line AD". As the 
radius vector continues its motion, the chord D,A becomes 
greater than a, and, if the length D,M, is taken equal to a, 
one has a point M, situated within the circle. Finally, as the 


CHAP. II. EXAMPLES. 29 


radius vector takes the direction AX’, the point M, falls in H. 
Thus the interior arc 4M,H is the prolongation of the exterior 
arc GM'A. The other half of the rotation gives the arc 
HNAN'G symmetrical to the first with respect to the line 
X'X, and completes the curve. 


27. Let us obtain now the equation of the curve in polar 
co-ordinates. Take the point A as pole, and the line AX 
as polar axis. Call w the angle which the radius vector 
makes with the direction AX. When the radius vector has 
the position of the line ADM, the right triangle ADB gives 
AD = b cos a, and, consequently, 


p= DM+ AD=a4+5 C08 w. 


When the prolongation of the radius vector intersects the 
circumference, as is the case in the position AD’, the angle o 
is the angle X.AD'; the right triangle BAD, gives 


D,A =— bcosw. 
and hence, 
p= DM, — D:A=a+ bcos w. 

But, if the radius vector, in Fig. 21, has the direction AD", 
the length is measured in the opposite direction to AD! 
Therefore, the radius vector of the point M will be AM, 
affected with the — sign; whence, one has 


p =— AM; = DM, — AD; = a + bos w. 
Thus, the entire curve is, in any case, represented by the 
equation 
(1) p=a4+)coso. 


If the point A is taken as origin, the diameter AB as axis 
of a, and a perpendicular to it at Aas the axis of y, then the 
equation of the curve in rectilinear co-ordinates will be 


(2) (a? + y? — ba)? = a? (a + y’). 
Equation (2) is derived from (1) by putting ~ for cosw and 


w+ y° for p? (§ 20), and squaring in order to remove the 
radical. 


30 PLANE GEOMETRY. BOOK I. 


28. The same curve may be obtained by another process. 
Being given a circle GH and a fixed point A, think of a mov- 
able tangent CM revolving on the circumference of the circle, 

and drop from the point A a per- 
; pendicular AM on this tangent 
Mi) Te (Fig. 22); find the locus of the 
point M. There will be three cases 
WZ ar to consider, according as the point 
aK i 7 a lies within, on, or without the cir- 
cumference of the circle. Suppose, 
Peng < for example, that the point A lies 
without the circumference. When 
the tangent touches the circumfer- 
ence at G, the perpendicular from 
A coincides with the diameter AG and the point G is a point 
of the locus. As the tangent revolves about the quadrant 
GCC', the point M describes the are GMM' of the curve. 
When the tangent descends to the position C'A, the point W 
describes the arc M'A. The tangent continuing its motion 
along CH, the foot of the perpendicular falls below the 
- diameter and describes the are ANH of the curve. The tan- 
gent has revolved about the semi-circumference GC'H; when 
the tangent revolves about the lower semi-circumference, the 
point M will trace a portion of the curve symmetrical to the 
first half. | 

In order to get the equation of this curve in polar co-ordi- 
nates, represent the radius of the given circle by-a, the dis- 
tance AB by b, and draw through the center B, of the circle, a 
line BD parallel to the tangent CM. Whence it follows 


p= AD+ DM=bdcosw+a. 


This equation is identical with equation (1) of § 26; therefore 
the curves which they represent are identical. 

Moreover it is easy to verify geometrically this identity. 
The angle D being a right angle, the locus of the point D is the 
circumference described on AB as a diameter. The point MW 
will therefore be obtained by prolonging the chord AD till DM 
is equal to BC. 











Fic. 22. 


CHAP. Il. EXAMPLES. — 31 


THE Rosse oF Four BRANCHES. 


29. Being given two lines OX and OY at right angles to 
each other, on which the extremities of a right line P@ of con- 
stant length are free tomove, 
find the locus of the foot of 
the perpendicular OM drawn 
from O to PQ (Fig. 23). 
When the line PQ coincides 
with OY, the point M coin- 
cides with O and the chord “A 
- OMtakes the direction OX; ” H 
therefore the tangent to the =| 
arc OM at O coincides with 
OX. The point £, the mid- 
point of PQ, describes a cir- 
cumference of which the cen- 
ter is O, and the radius equal 
to a; if the constant length 
be represented by 2a, the perpendicular OM is less than the 
oblique line OJ; therefore the distance OM is a maximum 
when the right line PQ is perpendicular to the bisector OA. 
As the movable line PQ continues its motion, it will assume a 
position P’Q! symmetrical to PQ with respect to the bisector 
OA, and one finds the are OM'A the symmetrique of the are 
OMA with respect to OA. The same curve is reproduced in 
each of the other right angles. Hence the curve has four axes, 
the two fixed right lines OX, OY, and the two bisectors A'‘A, 
B'B. The point O is the center of the curve. | 











Fig. 23. 


30. If the point O be taken as pole and OX as the polar 
axis, it follows from the right-angled triangles OMP, OPQ, . 


that 
p=OPcosw, OP=2asinw; therefore 


(1) | p= asin 2o. 


In rectangular co-ordinates the curve is represented by an 
equation of the sixth degree 


(2) (a? +’)? — 4 a?a’y’ = 0, 


$2 PLANE GEOMETRY. BOOK I. 


which follows at once from equation (1) by substituting re- 
ax 

spectively for cos w, sin w and p, ns ; and p= V 2 + 7 Squaring 

and transposing. : 


TANGENTS. 


31. The preceding examples show how to construct a curve 
from its geometric definition and to derive finally its equation. 
It is possible also in some cases to deduce from the geometric 
definition of a curve a simple construction of a tangent to it. 
Two remarkable examples will be given, —the curves described 
by the various points of a plane figure, which moves in a plane, 
and the locus of the feet of perpendiculars drawn from a fixed 
point to the tangents to a given curve. The construction be- 
longing to the first class of curves depends on the following 
proposition : | 


Lremma.— Every plane figure can be brought from one position 
to another in its plane by a rotation about a fixed point. 


It is first to be noted that the position of a plane figure in a 
plane is determined when one knows the position of two of its 
points. Let A and B be the two points of the figure in its first 
position (Fig. 24), A' and B' the same points in a second posi- 

tion; the line AB, of constant length, 
; is transferred to A'B'. Erect perpen- 
— ,  diculars to AA’, BB! at their mid- 
; points; the perpendiculars intersect in 
a point Z. The two triangles AJB, 
A'TB' are equal, since their sides are 
equal each to each, AB equal to A'B’, 
IA and JA! are equal, being oblique 
lines drawn from a point in a perpendicular cutting off equal 
distances from its foot, and similarly 7B and JB’ are equal; 
therefore the two angles AZB and A'IB' are equal; by sub- 
tracting the common angle A'JB, it follows that the angles 
ATA', BIB' are equal. Suppose now that the figure is revolved 
about the fixed point J, through the angle AJA’, the radius [A 
will fall on ZA’ and the point A on A’; in the same manner 








Ps 
Fig. 24, # 


CHAP. II. EXAMPLES. 33 


the line JB, revolving through the angle BIB’ equal to AIA’, 
will fall on JB' and the point Bon B’. Therefore this rotation 
about the point J brings the figure from its first position to the 
second. 


32. THEOREM. — Jf one considers the curves described by the 
different points A, B, C,-+-, of an invariable plane figure which 
moves in a plane, the normals to these curves, at points which 
correspond to the same position of the figure, intersect in the 
same point. 

Suppose A, B, C,---, to be different points of the figure in any 
position whatever, A', B’, C', ---, these same 
points in a new position. After what has 
been said, we can bring the figure from the 
first position to the second by revolving it 
about a certain. point 4; by this motion the 
lines L.A, LB, [,C---, describe angles respec- 
tively equal, and finally coincide with JA’, 
EB’, [,C',-»- (Fig. 25). The lines MQ, NE, 
PI,,++-, perpendicular to the ‘chords AA’, 
BB', CC", +++, at their mid-points, all intersect in the point J. 

Suppose now that the second position approaches continu- 
ally the first, and’ that the point 4, tends toward a limiting 
position J; the chords AA', BB', CC',---, prolonged, become 
tangents to the curves in A, B, C,---; the perpendiculars 
MI, NI, PI, ---, to the chords coincide with the perpendicu- 
lars to the tangents at A, B, C,---; that is, with the normals 
to the curves. Hence the normals to the curves described by 
A, B, +++, at these points all intersect in the same point J. 








Fig, 25. 


CoroLtuary. — If one could draw the normals to the curves 
described by the two points A and B of the movable figure, 
these two normals determine by their intersection the point 
I; by joining the point J to any third point C, one will have 
- anormal to the curve described by C; a perpendicular to the 
normal at C will be a tangent. This is the case if the two 
points describe straight lines or circumferences of circles. 
In the next section some applications of this method will 


be given. 
C 


34 PLANE GEOMETRY. BOOK I. 


33. In case two points of the movable figure describe right 
lines, it will be shown that the curve described by any other 
point is an ellipse; the preceding method will enable one to 
construct a tangent to the ellipse. 

Suppose, then, that the two extremities of a straight line 
CD of constant length lie on two lines OX, OY at right angles 
to each other, find the locus traced 














2 
by a point M of this line (Fig. 26). 
ene L When the line CD coincides with 
B ey on OX, the point M will fall on A at 
7A GEE as we a distance OA equal to DM; as the 
C. Sh : Se extremity D slides along OY start- 
Bae c f4 N° * ing from the point O, the point C 


approaches the point O, and the 
— point M describes the are AMB. 
When the line CD coincides with OY, the point M will fall 
on B at a distance MB equal to CM. The same arc is repro- 
duced in each of the four right angles, and the curve thus 
described is an ellipse. 

For, take the two fixed lines OX, OY as axes of co-ordinates, 
and call a and } the two constant lengths DM and CM, a and 
y the co-ordinates of the point M, then the similar triangles 
MPC, DQM give 

MP_CM ,, or Dee Ate ae _6 or t+t= 

DQ .DM Va — x pa 
which is equation (5) of § 13. Thus, the curve is an ellipse 
whose axes 2a and 2b coincide with the two giver rectangular 
AXES. 


34. It is not necessary that the point W be restricted to 
lying on the movable line between the points C and D; it can 
be situated on the prolongation. Consider the line C'D' of 
which the two points C' and D! slide on the two perpendicular 
lines OX and OY, and find the locus described by the point 
M. If a and bd be put for the distances D'M and C'M, the 
similar triangles MPO', D'QM will give, as in the preceding, 


MP MC! y b 


DQ. MD Jeug G 


CHAP. II. EXAMPLES. 35 


The construction of a small instrument called an edliptical 
compass depends upon this property. 

The two feet are placed on the points C and D, taken at 
wish on the line CD, and a pencil point at the point M; the 
two feet slide in grooves placed on the perpendicular lines OX 
and OY; the pencil point M describes by a continuous move- 
ment an ellipse. 

It is evident that a straight line is its own proper tangent. 
The points C and D of the movable plane describe the lines 
OX and OY; the perpendiculars CI and DI to these lines 
determine the point J through which pass the normals to the 
curves described by the various points of the movable figure 
for every position of this figure. The line 7M is therefore 
normal to the ellipse at M described by this point; the line 
drawn through M perpendicular to the normal to the ellipse 
at that point will be a tangent. 


35. Imagine two points H and F of the movable plane to 
slide on any two fixed straight lines OA and OB (Fig. 27). 
The perpendiculars to these lines at 
the points H and F determine the 
point of intersection J of the nor 
mals. The circle described on OF 
as a diameter passes through the 
points E and F; the line HF and 
the angle HOF being constant, the 
diameter of the circle is constant. 
Suppose that the circle is situated 
in the movable plane, and controlled euatal 
in its movement by the motion of the line HF; this circle 
will always pass through the point O; every point D of the 
circumference will describe a straight line OY, since the 
inscribed angle FOD, which corresponds to the constant are 
FD, is itself constant. 

Consider any point M of the movable plane; draw a line 
through this point and the center A of the circle; the 
two points, C and D, the extremities of the diameter MK, 
describe two perpendicular lines OX and OY; whence it fol- 








36 PLANE GEOMETRY. BOOK I. 


lows that the point M describes an ellipse whose axes are two 
times the distances CM and DM, and have the same directions 
as OX and OY. The line JM is normal to the ellipse at W. 


CONCHOID. 


36. Being given a point A and a 
straight line CC’, draw through the 
point A any secant AD, and, begin- 
ning at the point D where it meets 
the line CC’, lay off on either side a 
given length DM and DN; the locus 
of the points M and W is the con- 
choid (Fig. 28). It can easily be 
seen that this curve has two infi- 
nite branches, one on each side of 
the line CC’, and asymptotic to this 
line. The left branch will have dif- 
ferent forms according as the given 
o’ length DM is less, equal to, or greater 

Fig. 28. than the perpendicular drawn from 
A to the line CC". 

This curve belongs to the preceding category : one can regard, 
in fact, the line AD as revolving in the plane, in the following 
manner, one of its points D describes the line CC’, while the 
line itself passes through the point A, about which it revolves ; 
a point M of this line describes a branch of the conchaid. Con- 
sider the point of the movable line which is in A, when the 
line occupies the position AD; this point describes a branch 
of the conchoid passing through the point A and tangent to 
the line AD in this peint; the normal to this particular branch 
of the curve is the line AJ, perpendicular to AD. The normal 
to the curve described by the point D is the line DJ, perpen- 
dicular to the line CC'; by drawing a straight line from the 
point of intersection J of the two normals to the point M, 
one obtains the normal JM to the curve described by the - 
point M; the perpendicular to JM at M is a tangent to the 
curve. 











CHAP. II. EXAMPLES. 97 


The limacon (§ 26) is a curve analogous to the conchoid ; 
it is sufficient to replace the line CC’ on which the point D 
slides, by the circumference of a circle 
(Fig. 29). Consider then the point of 
the movable line which is in A, when 
the line occupies the position AD. This 
point describes a curve passing through 
the point A and tangent to the straight 
line AD at this point; the normal to this 
curve is the perpendicular AZ. The nor- 
mal to the circumference described by the 
point D is the diameter DI; the point of intersection J of the 
normals is therefore the extremity of the diameter which passes 
through the point D; the straight line JM is a normal to the 
curve described by the point M. 





37. The same construction is also applicable to the cissoid 
and strophoid; but it is necessary beforehand to give these 
curves another geometrical defi- 
nition. Consider a right angle . 
ABC (Fig. 30), of which a side ; g K 
BA passes through a fixed point 3 
A, and a point C on the other 
side slides on the line FE’; it is 
further supposed that the length 
BC is equal to the distance AO 
of the point A from the line 
EE'; the point M, mid-point of a a 
BC, describes a cissoid, and the Fig. 30. 
vertex B of the right angle a strophoid. 

In fact, the two right triangles ABC, AOC being equal, the 
angles CAL, ACL are equal, and the triangle ALC isosceles ; 
since AB is equal to CO, one has also LB = LO; therefore 
the locus of the point B is a strophoid (§ 28). 

The triangle ACP is also isosceles; join the point M to the 
mid-point D of AO and prolong this line till it is intersected 
in K by CK, drawn parallel to AO. The triangle CMIC being 
isosceles, it follows that Ch = CM= AD. Finally, describe 


E G 

















38 PLANE GEOMETRY. BOOK I. 


a circumference about the point O as a center with a radius 
OD; let F be the extremity of the diameter DO and H the 
point of intersection of the circumference with DK. The isos- 
celes triangle MCK is equal to DOH, DH= MK, whence 
DM = KH; moreover, the line FG is tangent to the circum- 
' ference at F. Therefore the locus of the point M is a cissoid 
having the point M for vertex and the line GG for an asymp- 
tote (§ 20). 

Consider now the point of the movable figure which is in 
A, when the right angle occupies the position ABC; this 
point describes a curve passing through the point A and tangent 
to the line AB at this point; the line AJ, perpendicular to AB, 
will be a normal to this curve. Further, the line CJ, perpen- 
dicular to EE’, is a normal to the curve described by the point 
C; the point of intersection J of the two normals is the common 
point of intersection of all the normals. Therefore the lines 
IB and IM are normals, the one to the strophoid, the other to 
the cissoid. 


PEDALS. 


38. The pedal of a given curve AB is the locus of the foot 
P of the perpendicular dropped from a fixed point O upon any 
line MP tangent to this curve (Fig. 31). A neighboring tan- 
gent M'P' will give a second point P! of the pedal. Let D be 
the point of intersection of these two tangents; the circle 

described on OD as a di- 
ae. ; ameter passes through the 








a Ss - : points P and P’, and the 
ra $4 s line PP' is a secant of 
; ap. Y the circle. Suppose now 

P “7 that the point M' ap- 
\l Z proaches indefinitely the 
bd Fig. 31. point M, the point D will 


ultimately coincide with ©, 
and the diameter OD with OM; the secant PP! will at the 
same time become tangent to the circle and to the pedal; the 
normal to the pedal will therefore coincide with the normal to 
the circle constructed on OM as a diameter, and this normal 


CHAP. II. EXAMPLES. 39 


may be found by joining the point P to the point C, the mid- 
point of OM. 

This construction may also be applied to the limacgon, which 
is the pedal of a circle (§ 28). But it is seen later (§ 307) 
that the construction of the tangents to pedals is reduced to 
the general method sketched in § 382. 


EXERCISES. 


1. A variable triangle ABC, whose vertex A is fixed and 
the angle A constant, is inscribed in a given circle. Show 
that the locus of the center of the circle inscribed in and 
escribed about the triangle is represented by two limacons. 

2. Show that the locus of the vertices of angles of given 
magnitude, whose sides are tangents to two given circles, is 
represented by two limacons. 
~~ 3. A variable circle touches a given. circle in a given point, 
and a tangent is drawn common to the two circles. Show that 
the locus of the point of contact of this tangent with the vari- 
able circle is a cissoid. 

4, A variable plane moves in a fixed plane in such a manner 
that two straight lines of the variable plane remain respectively 
tangent to two circles of the fixed plane. Show that a point 
on the fixed plane traces an ellipse on the movable plane. 

5. Construct the curves which, in the first system of bi-polar 
co-ordinates, are defined by the equation w+ nv=a. Show that 
of the three equations u+nv=a,u—nv=a, —u+nv=a, in 
which the two constants a and n have the same values, two 
alone define geometrical loci. These loci are closed curves, 
the one within the other; one calls them the conjugate ovals 
of Descartes. They are represented by the same integral alge- 
braic equations in rectangular co-ordinates. On the line which 
passes through the two poles there exists a third point, such 
that by taking this point and one of the first as poles, the 
equation preserves its form. 

6. If, being given two circles, any secant be drawn through 
a fixed point taken on the line of centers, and each center be 
joined to one of the points of intersection of the secant with 


40 PLANE GEOMETRY. | BOOK I. 


the circle, show that the point of intersection of these two 
lines describes the ovals of Descartes. 

7. The projection of the curve of intersection of two cones 
of revolution, whose axes are parallel to a plane perpendicular 
to the axes, is a system of the ovals of Descartes. 

8. Construct the curve which, in the first bi-polar system, 
is represented by the equation w+ v = a’, 2a being the distance 

between the two poles. This curve is called the lemniscate. 

9. Find the locus of the vertex of a triangle whose base a 
remains fixed, and in which the other two sides b, c, and the 
corresponding median m satisfies the relation b—¢c=m-~ V2 
(lemniscate). 

eS ine. straight line and a circumference each revolve with 
a uniform motion about a fixed point common to the two lines, 
the ratio m of the two angular velocities, is affected with the 
+ or — sign, according as the rotations are in the same or 
opposite sense; required to find the locus described by the 
second point of intersection of the two lines. 

Discuss the following particular cases: 


m= 4, or m= }, the limagon of Pascal; 
m= —1, or m=1, rose of four branches ; 


3 
m= —%, orm=4; 
m=2, or m=2. 
11. Solve the same problems, taking for the revolving curves 
two equal circumferences which revolve about a fixed point 
common to them. et 
Discuss the cases: 
m = 2, limacgon of Pascal; 
m = 3, rose of four branches ; 
m=—2; 
m= — 3. 


CHAP. III CONCERNING HOMOGENEITY. 41 


CHAPTER III* 
CONCERNING HOMOGENEITY. 


39, Derrnition. — The function f(a, b, ¢, +++) is said to be 
homogeneous with respect to the letters a, b, ¢, ---, when, on re- 
placing a by ka, b by kb, ---, one has 


St (ka, kb, ke, +++) = k"f(a, b, c, +++)5 


the exponent m being the degree of the homogeneous function. 
The following are examples of such functions: 


: avVb + dVe sin= 
a’ + 2ab, ray; : 
the degree of the first is 2, of the second 4, of the third 0, of 
the fourth —2. 
One can easily see: 
1° That the sum or difference of two homogeneous functions 
of the same degree is a function of the same degree as the given 
function ; 
2° That the product of several homogeneous functions of 
any degree whatever is a function whose degree is equal to the 
sum of the degrees of the given functions ; 
3° That the quotient of two homogeneous functions is a 
homogeneous function whose degree is equal to the excess of 
the degree of the dividend over that of the divisor; 
4° That the power of a homogeneous function is a homo- 
geneous function whose degree is equal to the degree of the 
given function times the exponent of the power; 
5° That the root of any homogeneous function is a homo- 
geneous function whose degree is equal to the degree of the 
given function divided by the index of the root; 


a+Vab a 
ate @406?’ 





42 PLANE GEOMETRY. BOOK I. 


6° That a transcendental function of a homogeneous func: 
tion of degree 0, is a homogeneous function, and of the degree 0. 
For example, the functions 


( ab } & + Va? + 4 
n{— ;|, log 
a’ + b* 








a+b 


are homogeneous and of the degree 0; because if a and b are 
replaced by ka and kb, the letter k disappears under the 
transcendental sign. But if the quantity placed under the 
transcendental sign, though homogeneous, were not of the de- 
gree 0, the letter k could not be removed from under the 
transcendental sign and the function would not be homoge- 
neous. 

Thus, the function sin (@ + /be) is not homogeneous, for 
here sin (ak + Vbck?) = sin (a + Vbe)k. 

When a monomial is rational and integral with respect to the 
letters a, b, ¢, +++, the degree of the monomial with respect to a 
letter will be the exponent of this letter in the monomial: the 
degree of the monomial with respect to several letters is the 
sum of the exponents of these letters. A monomial is always 
a homogeneous function, of a degree equal to the degree of the 
monomial; therefore the sum of several monomials of the same 
degree is a homogeneous polynomial of this same degree. For 
example, the polynomial 


a —4a°b+ 5ab? —2 0 


is a homogeneous function of the third degree, with respect to 
the letters a and 0. 


40. In seeking the relations which exist between the lengths 
of the various lines A, B, C, ---, of a figure, one thinks of these 
lines as being expressed in terms of a unit of length, which 
usually is not specified and remains to be chosen at will. Re- 
present by a, b, ¢, +++, the numbers which thus express the meas- 
ures of the lines of the figure and suppose that one has found 
between the numbers the relation 


(1) ST (a, 6, ¢, vo) = 0. 


CHAP. III. CONCERNING HOMOGENEITY. 43 


' The steps made in arriving at this result being independent of 
the unit of length, it is evident that this relation exists what- 
ever be the unit of length. Call «, 8, y, +++, the particular values 
of a, b, ¢, «++, for the first unit; @', B', y',---, the values of these 
same quantities for another unit; the two sets of numbers 
satisfy the relations 


(2) : St (@, B, Y vee) Uy 
(3) S(@, Bi yy +) = 0. 


But as the unit is changed the numbers vary proportionately, 
of the sort that, if k designate the ratio of the first unit to the 
second, one has 


_whence @=— he, B= hey — ky. 
If these values are substituted in the relation (3) it becomes 
(4) S (ka, kB, ky, +++) = 0. 


Consider that the first unit is fixed and the second varies; 
a, By y, +++, will be constant numbers and equation (4) will be 
satisfied whatever this number k may be. 

Thus, if equation (1) is satisfied when the letters a, b,c, +++, are 
replaced by a, B, y, ++, it will be satisfied when the letters are 
replaced by ka, kB, ky, +++, whatever the number k may be. 


41. The preceding condition is evidently fulfilled when the 
first member of equation (1) is a homogeneous function of the 
‘letters a, b, c, +++; because then one has 


S (Kee, KB, Key, +++) =k" f(a, By yy +++) 5 
if the expression f(a, B, y, ---) 1s zero, the same will be true of 
S(ka, kB, ky, +++) whatever k may be. 

Conversely, in order that the previous condition be fulfilled, 
it is necessary that the equation be homogeneous. The only 
case considered here is that in which the equation is algebraic. 

Suppose that f(a, b, ¢, ---) be an integral polynomial; if all 
the terms are not of the same degree, there will be groups of 


44 PLANE GEOMETRY. BOOK I. 


them which will be of the same degree; call ¢(a, 3, c, ---) the 
collection of all the terms of the degree m, the highest, 
Y(a, 6, ¢, +++) the collection of all terms of the next degree n, 
etc., the equation (4) becomes 


k™b(a, B, Y vee) a ky (ct, B, Y vee) + ee = 0. 


In order that this equation be verified, & being arbitrary, it is 
necessary that there exist separately 


b(@, B, y; --)==Q, (a, B, y, vee) =O, oes 


If the unit to which the numbers «, B, y, +++, are referred is 
arbitrary, then there must exist between the lines of the fig- 
ures the homogeneous relations 


p(a,.B; ¥,-°--) = .0;.. Wa, b;.¢, sto) == 0, 5:5) 


Therefore, if equation (1) is not homogeneous it is equivalent 
to several equations, separately homogeneous. 


42. It can happen that a homogeneous equation may be 
satisfied, where a particular unit has been chosen, without the 
parts which compose it being zero separately ; however, if the 
unit be changed, the equation will no longer be satisfied. 

This is illustrated by the example: Determine the dimen- 
sions of a cylinder whose total surface shall be equivalent to 
that of a sphere of radius A and its volume to that of a sphere 
of radius B. ¢ 

Let X be the radius and Y the height of the cylinder; call 
a, b, x, y, the measure of the lines A, B, X, Y, referred to any 
unit whatever; the unknown quantities will satisfy the two 
equations: 


(5) 227+ 2ay —4a’? = 0, 
(6) xy —40> = 0. 


Each of these equations is homogeneous; the one is of the 
second degree, the other is of the third. If they are satisfied 
when the lines are measured in a certain unit, the same will 

be true when they are referred to another unit. 


CHAP. Ill. CONCERNING HOMOGENEITY. 45 


The unknowns w and y satisfy also the non-homogeneous 
- equation 

(7) (20? +2ay — 4a”) + (ay — $ 5°) = 9, 

which is obtained by adding equations (5) and (6) member to 
member. 

Consider now equation (7), disregarding its origin. Four 
lines, A, B, X, Y, can be found such that, if they be measured 
in a particular unit, the numbers obtained verify this equation 
without annulling separately the two parts. Suppose, for 
example, that the lines referred to a first unit have for meas- 
ures the four members a=1, b= 3, e=2, y=4, of which 
three have been taken arbitrarily and the fourth determined 
_ by equation (7); if the lines are measured in a unit half as 
large, then one gets the numbers twice as large, a= 2.0 = '6, 
a—4, y=8, which do not satisfy the equation. The cyl- 
inder constructed with the lines X and Y thus determined 
enjoys the property, that the sum of the numbers, which, with 
the unit chosen, express the measures of its surface and of its 
volume, is equal to the sum of the numbers which express the 
measure of the surface of a sphere and of the volume of 
another sphere; but the same relation does not exist when the 
linear unit changes. Equation (7) can only be satisfied by the 
measures of the same lines when the unit of length is changed 
arbitrarily, provided these lines satisfy equations (5) and (6) 
taken singly. In the solution of problems of geometry, one 
never uses combinations of equations analogous to the pre- 
ceding. The equations which give immediately the theorems 
of elementary geometry, are homogeneous; and when equa- 
tions are added member to member it is to obtain a new equa- 
tion more simple than the proposed; for this it 1s necessary 
that the equations added be of the same degree. The prin- 
ciple of homogeneity serves in each instance to verify the 
algebraic transformation deduced. 


43. In case one of the lines of the figure is taken as the unit 
of length, the equations cease to be homogeneous; but it is 
easy to re-establish homogeneity. Let 


(8) F(b!, cl, -) =0 


46 PLANE GEOMETRY. BOOK I; 


be the equation which is obtained when a line is taken for | 
the unit; the letters b’, c', ---, represent the measures of the 
lines B, C, -+-, with respect to A. Choose an arbitrary unit, 
and call a, b, ¢, +++, the measures of the lines A, B, C, ---; then 
will 


se aro ry 
eb Fe 
b Cc 
whence by to, 
a a 


b ¢ 
© r(43--)=0, 
which is homogeneous. 


Thus, for example, if the sides of the right angle of a right 
triangle be referred to the hypotenuse of this triangle taken 
for the unit of measure, the measures of the sides satisfy the 
non-homogeneous equation 

b” +- cl? se a, 


from which is deduced the homogeneous equation 


b? Yd = ; A 
aa or bb +C=a, 


by replacing b' by : and c! by “. 


The curves, ellipse, hyperbola, parabola, cissoid, etc., studied 
in the preceding chapter, are represented by homogeneous 
equations. Any homogeneous equation 


I (@, Y, My b, C, vee) = 0, 


between the variable co-ordinates « and y of a point of the 
plane and the lengths a, b, c, --- of the various given lines, 
determine a curve, of which the position and dimensions are 
independent of the unit with which the lines are measured. 
Consider, on the contrary, a numerical equation in x and y, 


St (a, y) = 9; 


CHAP. Ill CONCERNING HOMOGENEITY. 47 


that is, an equation which does not involve other letters than 
a and y, and suppose their equation to be non-homogeneous. 
In order to represent by points of the plane the real solution 
of this equation, it is necessary to begin by choosing arbitrarily 
a scale, or the line to be employed as the unit. When the 
scale varies, the curve is no longer the same. It will be seen 
later that the various curves obtained in this manner have 
remarkable analogies; they are called homothetic curves. 


44. Remark I.—It frequently happens that one considers 
the numbers which represent the measures of lines, surfaces, 
and volumes. The units of surface and of volume, as well as 
the unit of length, remain indeterminate; but one habitually 
assumes that there exists among them this relation, that the 
unit of surface is the square constructed on the unit of length, 
and the unit of volume is the cube constructed on the same 
line. In this case, in order to verify the homogeneity of a rela- 
tion in which certain letters S and V represent the measure of 
a surface and a volume, these letters are replaced by p? and q’, 
where p and qg represent a side of the square and an edge of 
the cube equivalent to the surface and the volume considered. 
By this change the equation will contain only the lines. More- 
over, their substitution may be dispensed with, namely, in 
evaluating the degree of each term the exponent of a letter 
which designates a surface is doubled, and a letter representing 
a volume is tripled. 


Remark IIJ.—In general, when angles enter into a calcula- 
tion, these angles are referred to a unit definitely determined, 
and their measures are fixed numbers. In evaluating an angle, 
an arc of a circle is described about its vertex as a center with 
an arbitrary radius, and the ratio of this are to the radius is 
taken as the measure of the arc; the unit arc is the are which 
is equal to the radius. The trigonometric functions of angles 
are therefore numbers. In the application of the principle of 
homogeneity, one introduces the abstraction of letters which 
represent the angles or their trigonometric functions. 


48 PLANE GEOMETRY. BOOK I. 


CONSTRUCTION OF FORMULAS. 


45. In solving, if it be possible, the equations of a definite 
problem, one determines the formulas which represent the 
arithmetical operations that it is necessary to perform on these 
numbers which measure the known magnitudes in order to 
find the numerical values of the unknown. But can one not 
deduce from each formula, or what is the same from each 
equation, an appropriate graphical construction to give, not 
merely the numerical value of the unknown, but the unknown 
itself? In a word, is it possible to replace the numerical 
operations by graphical? In elementary geometry, the con- 
structions are considered which can be accomplished by means 
of a limited number of straight lines and circles, and which, 
consequently, can be made with the use of a rule and a com- 
pass. Since the circle is the most simple of curves and the. 
most easily constructed, the ancient geometers set a great price 
on this sort of construction; on the other hand, being ignorant 
of algebraic analysis, they did not have the means to decide if 
the questions which they had in view were susceptible of this 
kind of a solution, and it was not until they had made many 
fruitless efforts that they decided finally to investigate other 
curves. Their investigations have made certain problems 
celebrated which can be shown to-day not to be solvable by the 
straight line and circle. Examples of such are the duplication 
of the cube, the trisection of an angle, etc. 

The unknown quantity is assumed to be a straight line; 
when the unknown is a surface or a volume, it is represented 
by ax or a’x, a being a line taken arbitrarily; the construction 
of the line x gives a rectangle or a parallelopiped equivalent 
to the surface or volume sought. The determination of an 
angle given by one of its trigonometrical lines is reduced also 
to that of a straight line. It can be assumed then that every 
letter, such as 2, designates a straight line. 


46. Ratronan Formuta.—The formula which gives the 
unknown « ought to be homogeneous and of the first degree; 


CHAP. III. CONCERNING HOMOGENEITY. 49 


it can, however, be integral, rational, or irrational. When it is 
integral, it takes the form 


e=atb+ct+., 
and the length x is found by measuring one after the other, 
in one direction or in another, the lengths a, 0, ¢, ++. 
The fractional formula is the most simple: 
_ab 
=— 
The unknown is a fourth proportional, which can be con- 


structed by two parallels or by a circle. 
In the same manner may be constructed the formula 


ene & . b | ed 
a'b'e! q0 Cc 
by means of the series of fourth proportionals, 
cd b ap 
oY —- a B ae a C= cate 


By the aid of the preceding construction, a -monomial 
abe +++ ghi-+-- 

a'b'c! «+. q' 
ai---l, or, further, to the form A"~"t, A being any length and ¢ 
a line determined by the formula 





‘ of the degree m, may be reduced to the form 








Lee | 
t= Cau 
Consider now the formula 
ao. A—B+0C 
eA Ree Or 


in which A, B, C designate monomials of the degree m + 1, 
A', B', C' monomials of the degree m: each monomial may 
be reduced to the simple forms 


AG, A ey A A A es 
whence it follows that 


~AG@—b+ec)_ ,a 
ST ale be ar} 





50 PLANE GEOMETRY. BOOK I. 


That is, the unknown « can be determined by a fourth pro- 
portional between the lines £, «, X. 

If the fraction were of the degree m, the preceding opera- 
tions would reduce it to the form 


ye M yet 


47. IRRATIONAL FoRMULA OF THE SECOND DEGREE. — Let 
the formula in this case be 


=. a 2& 
P= Vad, or =o. 
ee 


The unknown # is amean proportional between the lines a and 
b; it is constructed by a right triangle, or by a tangent to a 
circle. When the quantity under the radical isa rational func- 
tion of the degree m, the formula is transformed as follows: 


m — 2 


Wt A 


Consider next an irrational formula of the second degree, in 
which the quantities are supposed to be connected by the + 
or — sign, are homogeneous, and of the same degree. For the 
sake of clearness, suppose that the value of x is reduced to the 
form 


c= D’ 
N and D representing functions in which the sign of division 
does not enter, neither do fractional nor negative exponents; it 
can also be assumed that neither the product of two radicals 
nor the product of a radical by an integral quantity enters the 
expression. In order to find the value of the numerator JN, it 
is necessary to perform certain operations in a definite order ; 
the first radical sign affects an integral expression, it will reduce 


m 


to the form Au; if this quantity be added to the others, they 
will be reduced to the same form, and consequently their sum 
also. A new radical sign may now be introduced affecting 


either an integral quantity, or a quantity with the exponent >) 


CHAP. III. CONCERNING HOMOGENEITY. 51 


m being odd. In every case, the radical will reduce to the 


form X‘v; this term is added to the others of the same form, 
and so on. Thus, it is seen that the numerator N will take 


the form Xt. The denominator can be discussed in the same 
manner. The unknown 2 being of the first degree, it can be 
found as a fourth proportional. 

One can demonstrate that the hypotheses which have been 
‘assumed in constructing the formula are necessary in order 
that it be homogeneous. 

Thus, every homogeneous expression of the first degree con- 
structed in any arbitrary manner by means of the symbols of the 
simple operations, addition, subtraction, multiplication, division, 
involution to an integral power, the extraction of a square root; i 
a word, every expression, rational and irrational, containing square 
roots only, can be constructed by means of a finite number of 
straight lines and circles. 

It can also be shown that only expressions of this sort-are 
susceptible of construction by the method just indicated; but 
this demonstration cannot appropriately be given here. For 
example, the edge w of a cube which is the double of another 
whose edge is a, is given by the formula 


a= V2a', 


and cannot be constructed by a rule and a compass. In like 
manner, it is, in general, true of roots of equations of the third 
and fourth degree, since cubical radicals enter in the expression 
of these roots. 


48. ConsTRUCTION OF THE Roots OF THE EQUATION OF THE 
Seconp Decree. —The equation of the second degree in one 
unknown quantity is reducible to the form 2°+pa+q=0; 
in order that it be homogeneous, it is necessary that the 
quantity p be of the first degree, and q of the second; whether 
these quantities be rational or irrational of the second degree, 
it will be possible to construct a straight line a equivalent to 
the first and a square b? equivalent to the second, and the 


52 PLANE GEOMETRY. BOOK I. 


equation of the second degree will assume one of the four 
following forms: 

“+ar+bh?=0, 

e+axr—b?=0, 

e—ax+h= 

e—ax—b?=0. 


The roots of the first and second equation are equal to those 
of the third and fourth, taken with contrary signs; it suffices, 
therefore, to consider those of the latter; if they be put under 
the form 

x(a — x) =0*, x(@—a) = bd, 
it is evident that it suffices to construct a rectangle equivalent 
to a square b*, and of which the sum or difference of the edges 
is equal to a given line a, problems which can be solved by 
elementary geometry. The solution of equations and the con- 
struction of formulas necessitate the discovery of theorems of 
geometry. 

The bi-quadratic equation may be reduced in a similar man- 
ner to one of the types 


xt + aba? — c’d? = 0, 
at — abe? + cd? = 0, 
x — abe’ — PWR =0; 
because it is useless to consider the equation at + aba? + cd? = 0, 


which has imaginary roots. If one put 2? = cz, these equations 


become- e 


eae a re 0, 2— ete = i ee 
c 


One solves these oe for z, then finds 2 by means of a 
mean proportional between c and z. 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 58 


CHAPTER IV 
TRANSFORMATION OF CO-ORDINATES. 


The equation of a curve in terms of certain co-ordinates 
being given, it is important to be able to deduce the equation 
of the same curve in terms of other co-ordinates. 

In order to discuss the problem in a general manner, it is 
necessary to deduce the formulas which express the co-ordi- 
nates of any point of the plane in a certain system in terms 
of the co-ordinates of the same point in another system. These 
formulas ate, moreover, useful in the investigation of a large 
number of other questions. 

First will be discussed the transformation of rectilinear 
co-ordinates of one kind into other rectilinear co-ordinates. 


TRANSPOSITION OF THE ORIGIN. 


49. Suppose that the two axes OX and OY be replaced by 
other axes O'X' and O'Y’, which are respectively parallel to the 
first (Fig. 832) and have the same direction. The position of 
the new axes will be determined by the co-ordinates a and 6 
of the new origin with respect to the 








primitive axes. Let x and y be the co- ee Ay 
ordinates of any point M of the plane ve 
with respect to the primitive axes; 2’ of $<; 
and y' the co-ordinates of. the same point i oe i, 

with respect to the new axes. Imagine the 7 ] x 
point O to be moved along the straight a 


line OM or the broken line OO'M to M, 
and project, parallel to OY, these two lines upon the axes OX. 
The projection of the line OM with the proper sign is the 
abscissa « of the point M; the projection of the line OO! is 
the abscissa a of the point O; the projection of the line O'M | 


54 PLANE GEOMETRY. BOOK I. 


on OX, or on the parallel axis O'X', is the new abscissa 2’. 
The projections of the two lines OM, OO'M, being equal, one 
hasw=a-+a!. By projection parallel to OX on the axis OY, 
one has in a similar manner y=b+y!. Thus are obtained the 
two relations, 


(1) e=a+2e,y=b+y/, 
between the old and the new co-ordinates of the point M. 


These relations are satisfied, whatever be the position of the 
point M in the plane. One may deduce from (1), 


(2) e'=a—a, y'=y—b. 


CHANGE IN THE DIRECTION OF THE AXES. 


50. Preserving the same origin, suppose now that the direc- 

tion of the axis is changed. Consider a particular case which 

has frequent application,—the case 

3! aes when the two axes are rectangular. 

- Suppose that the direction of the axes 

*" is changed by revolving the right 

= angle XOY (Fig. 33) through an angle 

= « about the origin till it attains the 

position X'O Y', and consider the angle 

| a as positive if the rotation takes place 

from OX toward OY, and negative if the rotation be accom- 
plished in an opposite direction. 

Through any point M of the plane draw MP and MP’ par- 
allel respectively to OY and OY’; let # and y be the co-ordi- 
nates of the point M with respect to the first axes, and a’ 
and y' the co-ordinates of the same point with respect to the 
new axes. ‘The projections of the two paths OPM, OP'M on 
any axis are equal. Project then these two paths on the axis 
OX; the projection of the length OP is the line itself, affected 
with the + or — sign, according as it is measured in the direc- 
tion OX, or in the opposite direction; that is, in every case, 
the abscissa x; PM being perpendicular to OX, its projection 
is zero; the projection of the first path reduces, therefore, to 2. 
Project now the path OP'M, projecting first the portion OP'; 








Fig. 33. 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 509 


if the length OP' is measured on OX’, it is necessary to mul- 
tiply by cos «, which gives for the projection OP' cos «; should 
this length be measured in an opposite direction, it 1s necessary 
to multiply by cos (#+«), which gives OP'- cos (m+ «@) or 
— OP'-cos«; but in the first case one has a! = OP’, and in 
the second a! = — OP': thus the projection of the line OP' is 
always expressed by x'cos a. Consider the second line P'M. 
If it be constructed in the direction OY’, it makes the angle 


ot ae with OX, and its projection is P'M- cos { «+ 5 ; if it be 
measured in an opposite direction, it makes the angle « + = +7 
with OX, and its projection is —P’M- cos/ «@ dee - but one has, 
in the first case y' = P'M, in the second, y'= — P'M; hence, 
the projection of P'M is always expressed by y' cos a+5 : 
Consequently, the projection of the path OP'M is always 
a! cos a + y' cos ( « +5), or x’ cosa —y'sine. By equating the 
projection of the two paths OPM, OP'M, one gets the relation 
x= «cosa — y'sin a. 

Project now the two paths on OY. The projection OP is 


zero; that of PM, affected with the proper sign, is y; thus the 
projection of the first path reduces to y. The two directions 


OX' and OY’ make respectively the angles — +aand+e 
with OY, which furnishes for the projection of the second 
path w' cos { — 5 -t «+ y' cosa, or w' sina+y'cos a, and one 
has the relation y = w' sina +y'cosa@. Therefore the formulas 
sought are 


(5) x= v'cosa—y'sina, y=w2'sina+ y'cos a, 


which express the old co-ordinates as functions of the new. 


51. Next the general question will be investigated. Let 
OX and OY be any two axes inclosing an angle 6, OX' and 
OY’, two new axes whose directions are defined by the angles 
a and £, which they make with OX (Fig. 34); one considers 


56 PLANE GEOMETRY. BOOK I, 


the angles @ and B as positive, when a movable straight line, 
starting from the position OX, gen- 

: erates them in revolving from OX 
toward OY, and as negative in case 
the line revolves in an opposite di- 
rection. From any point M of the 

| Ze. **" plane draw -the lines MP and MP' 


x 


! 
| 
| 
! 
! 
I 
! 
! 





Rey x respectively parallel to the axes OY 
ne and OY'. To get x, project the two 
oe paths OPM, OP'M on OH, perpen- 
ae HY « dicular to OY, so that a line, start- 


ing from the position OY, revolving 
in the direction OX through an angle equal to = will arrive 
finally in the position OH. Since the line OX makes with 
OH the angle 5a 6, and the direction OY is perpendicular to 


OH, the projection of the first path reduces to asin 6. The 
line OX' makes with OH an angle equal to the angle HOX 


increased by the angle XOX’, which together make 5 —6)+a. 


In the same manner the line OY' makes with OH an angle 


a 6\-+ 8; one has, therefore, for the projection of the sec 
ond path 
2'e0s(5—8+«) +y' cos(5-0+2) 
or w' sin (6 — w) + y'sin (6 — B), a 


which furnishes the relation 
# sin @= a' sin (6 — «) + y'sin (6 — £). 

To calculate y, project the two paths OPM, OP'M on a line 
OX perpendicular to OX, so that a straight line starting from 
OX and revolving through the angle 5 toward OY will coincide 
with Ok. Since the line OX is perpendicular to OX and the 
line OY makes with this line the angle — ; + 6, the projection 
of the first path reduces to ysin@. The angles which the lines 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 5T 


OX' and OY' form with OX are equal to the angles which 
they make with,OX diminished respectively by . which 


gives — 3 +a, and —5 +; the projection of the second 
path is therefore w! cos (— 5 = “) + y'cos (—5 -+ 8) 
or a sina + y'sin £, 
and one has the relation 
ysind=«2'sina+y'sin B. 
Thus are derived the formulas 


a! sin (@— a) +y'sin (6 — B) 
os : ’ 
sin 6 





(4) 








_ sine + y' sin B 
au sin 6 ( 
for the transformation of oblique co-ordinates into other 
oblique co-ordinates. 

It is a simple process to deduce the formulas serving to 
return from the new to the old co-ordinates. The angle 
between the new axes is B—a; the axes OX‘and OY form 
with OX'the angles — « and —a@+0; it suffices therefore to 
replace in the preceding formulas the angle @ by B—«, « by 
—a, B by 6 — a, which gives 











fr. vsin B + y sin (B — 8) 
: a sin (B — @) 
(5) ;_ — «sine +ysin (0—«) 

Keay sin (8 — @) 


Let the angle B — « between the new axes be represented by 
6'; then the determinant of the coefficients a’ and y’ in 
formulas (4) is 
sin Bsin (@—«) — sinasin(@—£)_ sin@! 
sin’ 6 ~ gin 8 





and the determinant of the coefficients # and y in formulas 
sin 6 


(5) is the reciprocal of the preceding, Aare 


58 PLANE GEOMETRY. BOOK I. 


52. The general formulas furnish certain special formulas 
which are of frequent use, 


1° The case when the primitive axes are rectangular. Here 6 


will be equal to e and formulas (4) will become 


(6) 


e=x' cosa + y'cos B, 
y=2'sina+ y'sin B. 


2° The case when the new axes are rectangular. Let B = a+ ., 
2 
then formulas (4) reduce to 


x’ sin (@ — a) — y' cos (6 — @) 
c= > ’ 
sin 0 





(7) ; 
; xe’ sina + y' Cos & 
a sin 0 








L 


One could also put B= a— > which would amount to 


changing the direction of the axis OY', and, consequently, the 
sign of y' in formulas (7). 


3° The case when the two systems of axes are rectangular. 
qc 


a one deduces formulas 


If, in formulas (6), one put B = @ + 
(3), already found, 


c= x'cosa—y'sina, 


(9) per 
y=usinae+y' cosa. 


t 


These formulas can also be derived by putting in formulas 
(= 
(1) 0=% 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 59 


GENERAL TRANSFORMATION. 


53. Suppose that the origin and the direction of the axes 
are changed at the same time. The new system of axes will 
be determined by the co-ordinates a and Y 
b of the new origin O', with respect to 
the old axes, and by the angles « and 
B which the new axes O'X' and O'Y' 
make with OX (Fig. 35). Through the 
point O' draw the two axes OX, and 7 : x 
OY, respectively parallel to OX and Fig. 35. 

OY. Then will in one case 


< 








G=aAth yootn; 


and in the other case, by virtue of formulas (4), 








a' sin (6 — a) + y'sin (0 — B) y _e#'sina+ y'sinB, 
i= 


Gey 73 
sin 0 sin 6 4 


substituting the values of a, and %, one general formulas of 
transformation become 


‘sin (6 — «) + y'sin (6 — B) 


Uy pemmtst 8? oe 
sin 6 





(8) 


x'sina + y'sinB 
sin 6 











aa 


The old co-ordinates « and y are expressed as linear integral 
functions of the first degree in the new co-ordinates a! and 7’. 


Tue TRANSFORMATION OF RECTILINEAR CO-ORDINATES 
INTO POLAR CO-ORDINATES. 


54. Let OX and OY be the rectangular y 
axes; take the origin as pole, and the a-axis 


as the polar axis (Fig. 36); by projecting 
the line OM on the axes OX and es one kL 


obtains the relations 








(9) L=pcosw, Y=p SIN. Fig. 36. 


60 PLANE GEOMETRY. BOOK I. 


Conversely, one can pass from polar co-ordinates to rec- 
tangular co-ordinates by means of the formulas 


p= VaF+ x, tan w = % 


Several transformations of this kind have been made, namely, 
when the equations of the cissoid, strophoid, limagon of Pascal, 
and rose (§§ 21, 24, 27, 30) were derived in rectangular co- 
ordinates. 


DISTANCE BETWEEN Two Pornts. 


55. Assume the axes to be rectangular and seek the distance 

of the origin from the point M, whose co-or- 

Y dinates are and y. From the right triangle 
OPM (Fig. 37) one has 


OM? = OP? + PM? = 2 4+ 7, 








0 ae 
Fig, 37. whatever be the position of the point M in 


the plane; whence it follows, by putting 
1 for the distance OM, 


(10) l=Ve¥+y7 


Seek, next, the distance between two points M and M’, situ- 
ated pyre in the plane; call # and y the co-ordinates of 
the point M, x! and y' those of the point M’' with respect to the 

rectangular axes OX, OY. Through the 

point M (Fig. 38) draw the axes MX’, 

u’ MY'parallel to the given axes. The co- 

‘ordinates of the point M' with respect to 

" the new axes are equal to w' — a, y' — y, by 

o x virtue of formulas (2) of § 49. The dis- 

hee tance of the new origin M from M' will 
therefore be, owing to formula (10), 


rt oy” 


re 
Mi 

















(11) l= V (a! — 2)? + (y' — y)*. 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 61 


56. In case the axes are oblique and the angle included 
by them is represented by 0, the ; 
expression will be somewhat more u 2 


complicated. a ; a) 
Seek now the distance of the origin — 





O from any point M of the plane. as : Ae 
In the triangle OPM (Fig. 39), what- x } i 
ever be the position of the point ™, : 
Fig. 39. 
one has 


OM’ = OP’ + PM’ —2.OP.- PM cos OPM. 


In case the point M is situated within the angle YOX, the co- 
ordinates x and y of this point are equal to + OP and + PM, 
and the angle OPM is the supplement of @; one has therefore 





(12) l= Var + 7? + 2ay cos 0. 
If the point M is situated within the angle Y’OX’, the co-ordi- 
nates « and y being equal to — OP and — PM, and the angle 
OPM, the supplement of 6, the same formula (12) is deduced. 
When the point ™ is situated within one of the angles YOX', 
Y'OX, the angle OPM is equal to 6, but one of the co-ordinates 
is positive and the other negative, which reproduces formula 
(12). This formula is, therefore, universal. 

In order to obtain the distance between two points Mand M’, 
one imagines, as above, axes drawn through the point parallel 
to the first, and obtains the formula 


(13) l=V(a'— av)" + (y'— yy +2 (e@'— 2x) y'—y) Cos 0. 





57. It is frequently useful to know the co-ordinates of a point 
which divides the distance between two given points in a given 
ratio. In case several segments are situated on the same line, 
one calls the direction of the segment the 7 
direction in which a movable point travels Ms 
that starts from the first point M and goes x 
toward the second M'. The algebraic 
value of the ratio of two segments is then 
the absolute value of their ratio, preceded ° x 
by the + or — sign, according as the two 2 








62 PLANE GEOMETRY. BOOK I. 


segments are measured in the same or opposite direction. Thus, 
MM, MM 
M.M" M,M"’ 








in Fig. 40, the ratio a is positive, the ratios 


are negative. 

Being given two points Mand M', on an indefinite straight 
line, having the co-ordinates x and y, w' and y’, find on this line 
a point M, with the co-ordinates 2, and y,, such that the ratio 
eo has in magnitude and sign the value mt If the axes are 
M,M' m 
transferred parallel to themselves to the point M4, the new 
co-ordinates of the points Mand M' will be «— a, and y— », 





! 
. Mey (eee . 
x'— x, and y'—y,. In case the given ratio — is negative, the 
m 


point sought, MW, ought to le between Mand M'; this is the 
case in the figure. The differences x — a, and x’ — a, or y—y, 
and y' — y, have opposite signs; their ratio is negative, and the 
absolute value of their ratio is equal to the absolute value of 





! 
oe . OF a One has, therefore, in magnitude and in sign 
o m 
(14) fe 





w— a y'—yY mM 
f ~ 
When = is positive, the point sought, Mj, lies without the seg- 


ment MM'; the differences 7—a, and w' —a, or y—y, and 
y' — y, have the same sign; their ratio is plus and equal to the 





! 
, or to ™'. Therefore equations (14) are‘also appli- 
l m 


cable to this case. Whence one has the following formulas 
m! 


which solve the problem for every value of the ratio —, 
m 


_ my — m'y' 
m — m! 


REMARK. — The co-ordinates of the point Mj, x, and y%, may 
be deduced from the preceding by changing the sign of m'. 


— me+ mie! 


) = my tmy, 
m +m! 


XH 
m+ m! 


CHAP. IV. TRANSFORMATION OF CO-ORDINATES. 68 


From the position of this point it follows that 
MM _— MM _m'. 


MM MM Mm’ 


the points M, and ™M,, corresponding to values, equal and 








! 
° : . - Mm . ° 
opposite in sign, of the ratio —, are called harmonic conjugates 
m 


! 

5 : a) 
with respect to the segment MM". Incase the given ratio — 
m 


is equal to —1, the point M, will bisect the segment MM’ and 
has the co-ordinates . 
aw + x! yty' 
me 7 Ue nae + i } 
the point M, is removed to infinity. 


r= 





! 
If ” is put equal to —X, it follows at once that the two 
m 
conjugate points have respectively the co-ordinates 


at ! 
ee Sieh es iain 








ae ee 
a, — tA! y y — dy! 
a 


CLASSIFICATION OF PLANE CURVES. 


58. Rectilinear co-ordinates are especially adapted to the 
study of the general properties of plane curves. In this 
system plane curves are classified in the following manner: 
They are distinguished as algebraic and transcendental, accord- 
ing as the equations which represent them are algebraic or 
transcendental. An equation is said to be algebraic when the 
co-ordinates « and y enter affected only by the symbols of 
algebraic operations. If, however, one of the co-ordinates 
enters affected by a transcendental symbol, as a sin, logarithm, 
tan, etc., the equation is said to be transcendental. Algebraic 
equations can always be put under an integral form by remov- 
ing the radicals and the denominators. 

One classifies algebraic curves according to the degree of 
their equations. Curves of the first degree (straight lines) are 


64 PLANE GEOMETRY. BOOK TI. 


those which are represented by equations of the first degree in 
x and y; the equation of the second degree furnishes curves 
of the second degree, ete. : 

It is very plain that the degree of any curve remains 
unaltered whatever may be the position of the axes of co-ordi- 
nates in the plane. In fact, let f(a, y¥) = 0 be the equation of a 
curve referred to certain axes OX and OY, m the degree of 
this equation supposed to be integral. To refer this curve 
to other axes O'X' and O'Y', it is necessary to substitute for 
x and y in the proposed equation the values given by the 
formulas of transformation (8); these formulas being of 
the first degree in the co-ordinates a’ and y', it is impossible 
that the equation in w' and y' be of a degree greater than m. 
The equation will not be of a degree less, because in that case 
the inverse transformation would increase the degree, which is 
impossible. Thus, the new equation is of the same degree as 
the primitive. 

The degree of a curve is the same as the number of points 
of its intersection with a straight line. In fact, let m be the 
degree of a curve whose equation is f(a, y¥)=0 when the 
straight line has been chosen as the a-axis; if in this equa- 
tion one makes y =0, the equation thus obtained in x will give 
the abscissas of the points common to the curve and the 
w-axis. Since the first member of the equation is not identi- 
cally zero, and is at most of the degree m, the equation cannot 
have more than m roots, and consequently the line has at most 
m points in common with the line. If the equation were 
satisfied by more than m values of x, the first member would 
be identically zero, and consequently the line would be a part 
of the locus; in this case, the polynomial f(a, y) vanishing 
identically when y is put equal to zero would contain y as a 
factor, and the equation f(a, y) =0 could be decomposed into 
two equations, one y= 0 of the first degree, the other of the 
degree m—1. 

Accordingly, curves of the first degree cannot be cut by a 
straight line in more than one point; therefore the curves are 
straight lines. Curves of the second degree cannot be cut by 
a straight line in more than two points; those of the third 


CHAP. Iv. JSRANSFORMATION OF CO-ORDINATES. 69 


degree, in more than three points. The circle, ellipse, hyper- 
bola, and the parabola are curves of the second degree (§$ 10, 
13, 15, 18). These curves can be cut by a straight line in 
two points. The cissoid and strophoid (§§ 21 and 24) are of 
the third degree. They can be cut in three points by a straight 
line. The limacon of Pascal ($ 27) is of the fourth degree; 
the rose of four branches (§ 30) is of the sixth degree. 

First, one studies curves of the first, then those of the 
second, and finally those of any degree whatever. 

When an algebraie integral equation of the degree m is said 
to represent a curve of the degree m, it is assumed that the 
first member cannot be decomposed into integral factors ; other- 
wise the equation could represent two or a greater number of 
curves of lower degrees. Thus, for example, an equation of the 
second degree, whose first member is the product of two inte- 
gral factors of the first degree, represents two lines of the first 
degree; that is, two straight lines. Similarly, an’ equation of 
the third degree may represent three straight lines, or one 
curve of the second degree and a straight line. It is for this 
reason that certain properties of curves of the mth order are. 
applicable to a system of m straight lines; that is, to a polygon 
of m sides. Thus is learned that the properties of curves of 
the second degree are applicable to a system of two straight 
lines, since this system can be considered as a locus of the 
second degree. | 

E 





Book II 


STRAIGHT LINE AND CIRCLE 


CHAPTER I 
STRAIGHT LINE. 


CONSTRUCTION OF THE EQUATION OF THE FIRST 
DEGREE. 


59. The general equation of the first degree between two 
variables 2 and y has the form 


(1) Ax + By + C=0. 


It has already been noticed that the line represented by this 
equation cannot be cut by a straight line in more than one 
finite point, and is necessarily straight. y 
However, it is best to show directly that 
this equation represents a straight line. : 








G B ee; 
It is impossible that the coefficients A and 
B be zero at the same time, for then C / 
must also be zero, and the equation is ai = 


reduced to an identity. But it is possible me. 2 
that one of the coefficients be zero. If, for example, the coeffi- 
cient A be zero, the equation takes the form By+ C=O, 


Y, 
‘whence y= af =b. This equation represents the locus of a 


point M whose ordinate is constant and equal to b, whatever 


the abscissa may be; the locus is a straight line parallel to the 
67 


68 PLANE GEOMETRY. BOOK IL 


axis OX (Fig. 41). This line is constructed by laying off on 
OY, beginning at the origin, a length equal in ahsolute value 
to b, in one direction or the opposite, according to the sign 
of 6, then drawing GG' through the point B parallel to the 
axis OX. Asa special case, the equation y = 0 represents the 
axis OX. 

When the coefficient B is equal to zero, the equation reduces 


to Ax+C=0, or e=— “ =a. This equation represents 


the locus of the point MW, whose abscissa is constant and equal 
to a, whatever the ordinate may be. It isa 
straight line HH" parallel to the axis OY 
(Fig. 42). This line can be constructed by 
laying off on the axis OX, beginning at the 
origin, a length OA equal to the absolute 
Bt is xX value of a, in one direction or the opposite, 
i according to the sign of a, then drawing HH' 
See through the point A parallel to OY. As a 
special case, the equation « = 0 represents the axis OY. 
In case the coefficient B is not zero, all the terms of the 
equation can be divided by B and it may be written 


z H 





20 
B B 
or (2) y = ax +b, 
by putting, for brevity, a = — < NE hss _ i, 


Consider next the particular case when b = 0. 


The equation then reduces to the form 


y 
Y= av, Or —~-=—= a4. 
y = ax, or > 


If a be a positive number, every point of the locus, having 
co-ordinates with the same sign, lies in the angle YOX or its 
vertically opposite (Fig. 43). Take an arbitrary abscissa OP, 
and draw through the point Pa line parallel to the axis of y; 
MP 


if a point M can be found on this parallel, such that OP > a, 


CHAP. I. STRAIGHT LINE AND CIRCLE. 69 


it will be a point of the locus. 
Let M, M', M", ---, be points 
of the locus constructed by the 
preceding rule; it follows from 
the equal ratios 


MP M'P' —M'P'" x’ 
_—_— — = —— 96 «Op 
OF . OF... OF! i 


that the triangles OPM, OP'M", 
OP"M",.--, are similar, and 
hence the angles MOP, M'OP’, 
M"OP", ---, are equal; there- Fig. 43. 

fore the points M, M', M",---, all lie on the straight line A’A 
passing through the origin. If # varies continuously from 
—c to+o, the point M will move 
continuously and describe an indefi- 
nite straight line AA’. 

When a is negative, all points of 
the locus, having co-ordinates of — 
opposite signs, le in the angles 
YOX' and Y'OX (Fig. 44). Let ©, 
M', M",---, be different points of the 
locus; then, as above, it follows from 
the relations 


MP MP! —M"P"_ 























= a, 


BOP. 20P 20ORa : 


that all these points are on the same straight line A'A passing 
through the origin. Thus, in every case, the equation y = ax 
represents a straight line A'A passing through the origin. 

Let us return now to the equation y=axv+b. By compar- 
ing the two equations y=ax+b, y=az, one sees that the 
ordinates corresponding to the same abscissa differ by a con- 
stant b; if therefore the ordinates of all the points of the 
straight line A'A are increased or diminished according to 
the sign of b by the lengths MN, M'N', M'"'N", ---, equal to the 
absolute value of 6 (Fig. 43), the points N, N', N”, .--, thus 
obtained, form evidently the right line B'B parallel to A'A. 


70 PLANE GEOMETRY. BOOK It. 


It follows from what precedes that any equation of the first 
degree between two variables x and y represents a straight line. 


60. It can be shown, reciprocally, that any straight line is 
represented by an equatin of the first degree. If the straight 
line be parallel to the axis OX, then all of its points have the 
same ordinate, and the equation has the form y = b (Fig. 41). 
If it be parallel to the axis OY, all of the points have the 
same abscissa and the equation will have the form w=a 
(Fig. 42). In case the straight line passes through the origin, 
it occupies one or the other of the two positions indicated in 
the figures 43 and 44, and the similar triangles give 





MP - M'P' me — M"'p" _ a 
OP OPE Pl. 
MP MP! —M'P" 

or =—_ — an 010.08 





OP 2 OP. OP! 


If a be this constant ratio, the equation of the right line is 


/—a, or y=aa. Suppose, finally, that the straight line is 
a 


not parallel to either of the axes nor passes through the origin 
(Fig. 45); according to what precedes, a line drawn through 
the origin parallel to this straight line will have the equation 
y = ax; now the excess of the ordinate of a point on the pro- 
posed line over the ordinate of the corresponding point on the 
parallel is a constant quantity 6; therefore the proposed straight 
line has’ for its equation y = ax + b. 


v 


MEANING OF THE COEFFICIENTS. 


61. The equation of every straight line which is not parallel 
to the axis of y can be put in the form 
(2) y=ax+ b. 

The constant } is the ordinate of the point H (Fig. 43) 
where the straight line cuts the axis of y; it is called the 
ordinate of the origin. 

The constant a determines the direction of the line; it is 
the same for all parallel straight lines and is called the angular 
coefficient or coefficient of direction. 


CHAP. I. STRAIGHT LINE AND CIRCLE. 11 


Draw through the origin a line A'OA parallel to the proposed 
straight line and situated with respect to the axis XX’ on the 
same side as the line OY. Let 6 be the angle XOY, a@ the 
angle AOQX, an angle which can vary from 0 to 7; it follows 
from Fig. 43 that 


ga ¥ MP _ sin MOP _ sin « 
a 


OP sinOMP sin(@@—a) 
and from Fig. 44 that 


roa eo MP___ sinMOP _ sin(r—@) _-_ sme. 
—~z —OP —sinOMP -—sin(a—6) sin(@—«)’ 


one has, therefore, in every case, 











ue, eee 
@) sin(0@—«) 7% 


If the axes are rectangular, this relation reduces to 
(4) tan «= a, 


and determines the angle « which OA makes with the axis OX. 
When the axes are oblique, one deduces from the relation 
(3) the formula 


sin « = asin 6cos « — acos 6 sin a, 





t EL ae 
oF. ©) ae 1 + acos 6 


In order that this formula may be solved by logarithms, the 
following transformation is made. It follows from (3), 


t ped 
a—1_ sina —sin(0—«@) ae an 3) 





a+i1 sina+sin(@—«a) ~ tan? 
or (6) tan (« aos 3) — a— tan °. 


62. In constructing the straight line represented by the 
equation of the first degree, with numerical coefficients, one 
usually seeks the points in which the straight line cuts the 
axes and draws a straight line through them. 


72 PLANE GEOMETRY. BOOK It. 


Suppose that the equation 2@—3y—=5 be given; for y=0, 
one has «=8; for r=0, y= — 33 starting from the origin, 
one lays off on the x-axis the length 3 in the direction OX, on 
the y-axis the length 8 in the direction OY'; through these 
two points the line is to be drawn. If the equation be free 
from an additive constant, the straight line passes through 
the origin. One determines then a second point, by giving 
to # a particular value; let, for example, 2y+32=0; the 
equation being satisfied for «=0, y=0, the line passing 
through the origin; if one makes a = 2, then one has y= — 8; 
construct the point whose co-ordinates are a = 2, y = — 9, and 
draw a line through it to the origin. , 

Pane 
63. The general equation of the straight line, 0” 


Az + By+C=0, 


contains but two arbitrary coefficients or parameters; because 
one can divide the equation by one of the coefficients, then the 
other two will be replaced by their ratios to the divisor. When 
the equation is put under the form y = ax + b, the two param- 
eters are @ and >. In order to fix the position of the straight 
line in'the plane, it will be necessary to give a value to each of . 
the two parameters or to be given two relations between them. 


64. Prosiem I.— To find the general equation of straight 
lines which pass through a given point. 
Let w' and y' be the co-ordinates of the given point M. The 
equation of any line is 
y= axr+b. 


If this line pass through the given point M, the co-ordinates 
of this point must satisfy the equation to the line; if therefore 
the variable co-ordinates « and y are replaced by the co-ordi- 
nates w' and y' of the point M, one will have the equation of 
condition, 

y' = au'+ 0. 


This relation between the two parameters a and b determines | 
one of them as a function of the other; for example, the pa- 
rameter 6 as a function of a. By replacing b in the equation 


CHAP. I. STRAIGHT LINE AND CIRCLE. 73 


of the straight line by its value y'— az! deduced from the 
equation of condition, one obtains the equation 


(7) y—y =a(e—x'). 


Equation (7), in which the angular coefficient a is arbitrary, 
represents all the straight lines which pass through the point 
M. When the parameter a is varied, the line revolves about 
the point M. 

It has been assumed that every straight line is represented 
by an equation of the form y=ax+b, whatever its position 
in the plane may be. But there is one exception, viz., when 
the straight line is parallel to the y-axis; because, in this case, 
the angular coefficient a is infinite, and the ordinate at the 
origin is &. Accordingly, if in equation (7) @ is replaced by 


the ratio , the equation may be put under the form 
n 
n(ly—y')=me—a'); 


and letting n = 0, one gets the equation « = 2’, which represents 
a straight line, drawn through the point M, parallel to the y-axis. 


65. Prosiem II.— Through a given point draw a straight 
line parallel to a given straight line. 

Let y=ax+0 be the equation of the given straight line 
AB, «' and y', the co-ordinates of the 
given point M (Fig. 45). Since the line 
is to pass through the given point M, its 
equation, as we have seen above, will 
have the form 


y—y=al(ae—x'). = Jo Xx 
This line will be parallel to the line AB seine 
when the angular coefficient a’ is equal to the angular coeffi- 


cient of the line AB. One will have, therefore, a’=a, and 
the parallel required will have for its equation 


eo 4 D 








y—y'=a(@—e'). 


66. Prosiem III.— Draw a straight line through two given 
points.. 


LA PLANE GEOMETRY. BOOK If. 


Let M and M' (Fig. 46) be the two given points, 2’ and y! 
the co-ordinates of M, x" and y" those 
of M'. The line MM’, passing through 
the point M, is represented by an equa- 
tion of the form 








Se (- x y—y'=a(a—2). 


Hig. &- It is a simple matter to determine the 


coefficient a so that this line may pass through the point M'. 
For this, it is necessary that the co-ordinates of the point M!' 
satisfy equation (7), which gives the relation 


y!! ae y! —a (a! pad x’), 


set 

whence one deduces a=2—~. 

oe! — x 
Thus, the angular coefficient of the line MM'is equal to the 
ratio of the difference of the ordinates to the difference of the 
apscissas of the two given points. If in equation (7) a be 
replaced by its value, one obtains the equation of the line MM", 


! 
(8) y-y'= LoL ea), 


an equation which can be written in the form 





FT RES Te = c 





When the point Mis at the origin, one has ae U, ¥ = 0, 
and equation (8) reduces to 


67. It is sometimes useful to define a line by the points 


Y where it cuts the axes (Fig. 47). Call a 
the abscissa of the first point, b the ordi- 
= nate of the second, and let 
b Ax + By+ C=0 





makes successively y=0 and «=0, one 


r a NG + be the equation of the line sought. If one 
Fig. 47. obtains the points where the line cuts the 


CHAP. I. STRAIGHT LINE AND CIRCLE. 15 


axes; one has ees b aes whence A = — & poe, 
A B a b 
By replacing A and B by their values, the equation takes the 


simple form, 


9 pce a ode 
(9) | nie 


68. Prostem IV.— Find the point of intersection of two 
given lines. 


Let Ax + By +C =0, 
A'e + Bly + C'=0, 


be the equations of the two given lines AB and CD (Fig. 48), 
M the point of intersection of these two 
lines. The point M being common to 
each of the two lines, its co-ordinates will 
satisfy at the same time the two equa- 
tions; if, therefore, one solves these two 
simultaneous equations for the two un- 
known quantities, 2 and y, we obtain the 
co-ordinates of the point M, 


_BC'-CB | _CA'~ AC’ 
AB = BA 9 = Ap BA 




















When the denominator AB'— BA!' is different from zero, the 
formulas furnish finite and determinate values for x and y, 
and the two lines intersect in a finite point M@. But when the 
denominator is zero and the numerators different from zero, 
the values of a2 and y are infinite; in this case, the two lines 
are parallel, and, in fact, they have equal angular coefficients 
—~A__A’ te one has *=2=S the two numerators 
B B! a Bp. 
and denominators will be zero at the same time, and the values 


of x and y will take the form 5 5 the intersection will be inde- 


terminate, and, in fact, the two proposed lines coincide; because 
! ! ' 
if one puts f=5=5=K then 3. A’ — AK. DP Ba, 


76 PLANE GEOMETRY. BOOK II. 


C' = CK; substitute these values in the second equation, and 
divide by J, the resulting equation will be identical with the 
first. 


69. ProsLEM V.— To find the general equation of a straight 
line which passes through the point of intersection of two given 
straight lines. Let 
(10) Ax + By +C =0, 

(tf) A'e + Bly + C'=0, 

be the equations of two given straight lines. One could first 
find the point of intersection of the proposed lnes by solving 
equations (10) and (11); then find the equation to any line 
through this point (§ 64). But one can arrive at the same 
result in a more rapid manner. 

If one multiplies equation (11) by an arbitrary quantity, 
then adds it member by member to equation (10), one gets an 
equation of the first degree, 

(12) (Av + By + C)+A(A'e + Bly + C') = 0, 

which represents a third line passing through the point of 
intersection of the first two; for, in fact, the co-ordinates 
of this point satisfy the two equations (10) and (11), annulling 
the two quantities put in parentheses, and consequently, satisfy 
equation (12). This equation (12), in which the coefficient » 
is arbitrary, represents any straight line which passes through 
the point of intersection of the two given lines; because one 
can determine this coefficient A so that the line may pass 
through any point M of the plane having as co-ordinates 
z' and y'; for this it suffices that the equation of condition, 


(Ax! + By! + OC) +A(A'2! + Bly' + C) =0, 


be satisfied, which gives 
_— Av'+ By + 
A'g! + Bly'4+ el 





(13) h= 


In case one makes A = 0, the equation (12) becomes 


Ax + By+C=0; 


CHAP. I. STRAIGHT LINE AND CIRCLE. i 


this is the first straight line. If one replace A by = after hav- 
ing multiplied by n, place n=0, one gets the second line, 
A'z + Bly+ C'=0. 
If in equation (12) one replace A by the value (13), one gets 
the equation 
(14) Ac+ By+C _Alv+ Byt+C 
Ag! + By'+C” A'z'+ Bly'+ Cr 








which represents the line passing through the point M and 
the point of intersection of the two given lines. The numera- 
tors are the first members of the given lines, the denominators 
are these same polynomials when # and y are replaced by the 
co-ordinates of the given point. One recognizes at once, from 
inspecting this equation, that the line it represents passes 
through the given point and through the point of intersection 
of the two given lines. 

When the two lines (10) and (11) are parallel, equation (12) 
represents all the lines parallel to them. 

An equation of the first degree in # and y, which contains 
an arbitrary constant A, represents an infinity of lines; when 
this parameter appears in the first degree in the equation, one 
can put the equation in the form (12); one sees then that all 
the lines pass through the same point, —the point of intersec- 
tion of the lines (10) and (11). 


Remark. — Suppose that four concurrent lines d, d', d,, and 
d, are given; then the lines d, and d, are called harmonic con- 
jugates of the lines d and d', when the two points where a 
secant cuts the lines d, and d, are harmonic conjugates of 
the two points where it cuts d and d' (§ 57). It is easy to 
see then that the two lines d, and d,, whose equations are: 


(d;) Ax+ By + C+A(A'e4+ Bly + C') =), 

(d,) Auw+ By+ C—d(A'a + Bly + C') =9, 
are harmonic conjugates of the given lines (10) and (11). In 
fact, cut the three lines (10), (11), and (12) by a secant hav- 


ing the equation y= ma+n and meeting these lines in the 
points M, M', and M,. 


78 PLANE GEOMETRY. BOOK II. 


The abscissas of the three points are 
—_ Pati? Bat Bat Cr NG at 0} 
A+ Bm’ A'+ Bim" A+ Bm+2(A'+ B'my’ 


and, according to the formulas of § 57, one has in magnitude 
and in sign, 





MM tee) a Bin 
MM x'—2, A+ Bn- 


Similarly, calling M, the point where the same secant cuts 
the line d,, 


—A 








MoM _ r A'+ B'm 

MM = A+ Bm’ 
as one can easily see by changing A into —r. Therefore, 
finally, 





MM __ MM. 

M,M'  M,M"’ 
which shows that the two points M,, M; are harmonic conju- 
gates of the points M, M' (§ 57). 








70. Prospiem VI.— The condition that three lines pass 
through a point. 


Let Ax +By +C =0, 
A'le + By +C' =0; 
A" + B'y + C"=0, 


be the equations of three given lines. One finds'‘the point 
of intersection of the first two lines, and substitutes the 
co-ordinates of this point in the third equation. This fur- 
nishes the equation of condition 


A" (BC' — CB') + B" (CA'— AC") + C" (AB'—.A'B) = 0, 
or C''(AB'—A'B) +C' (A"B—AB") + C(A'B"—A"B')= 0. 
The lines will not only intersect, but also be parallel if 

AB'— A'B, A'B"'— A"B', A"B— AB" 


are all three zero. 


CHAP. I. STRAIGHT LINE AND CIRCLE. 79 


Otherwise, the general equation of the lines which pass 
through the point of intersection of the first two is 
(Az + By + C)+A (Ae + By + C')=0 
or (A+ AA')\e+ (B+ rA(B) y + (C+ AC") = 0 
If the lines have a common point, by assigning a suitable value 


to A, this equation will represent the third line; therefore we 
ought to have 


A+\A'= KA", B+\B'= KB", C+\C'= KC", 


where £& is arbitrary, 
A+,XA' B+)B' C+)dC' 
wea PAU eu 


By eliminating A, one gets the equation already obtained. 





71. Exampie. —Consider the three medians of a triangle OAB 
(Fig. 49); take the vertex O as origin, the two 
sides OA and OB as co-ordinate axes, and 
designate by a and 6 the two lengths OA 
and OB. The median AF, cutting the axes 


at the distances a and : from the origin, has 


for its equation, 








x 
b 
similarly, the median BF has for its equation, 
2 oe 
mn 
a a8 b 


The mid-point of AB has the co-ordinates OF = 7 OF =e the line 


OD, which joins the origin and this point, has the sqinice, 


pte 
a 


By solving the first two equations, one gets the co-ordinates 


a b 
2 3° y= 3" 
of the point C the intersection of AZ and BF. These co-ordinates satisfy 
the third equation ; hence the third median OD passes through the point C. 
By applying the second method, we see at once that the three medians 
pass through the same point; for, by subtracting the second equation, 
member by member, from the first, we get the third equation. 


80 PLANE GEOMETRY. BOOK It. 


72. Prosiem VII.— Find the condition that three points lie on a 
straight line. 

Let x and y’, x!’ and y'’, «'” and y!'’, be the co-ordinates of three 
given points M’, M’’, M''’. If the points lie on 
a line, the preceding pairs of co-ordinates satisfy 
the equation Ax + By + C=0, and the deter- 
minant 

x, eee 
Pe ook 


al, yl, 1 








Fi ° 5 e . 
.> is Zero. 


The lines M’M'', M'M’'', coincide, and their angular coefficients are 
equal, then will 





yl! —y! oe ee por y! 
gil pate a! gl pba! ge! 


73. Examp.Le. — If the four sides of a quadrilateral OACB are pro- 
longed (Fig. 51), a complete quadrilateral OACBA'B' is formed; the 
sides intersect two by two in six 
points or vertices ; by joining the 
vertices one obtains the diagonals 
AB, A'B', OC; it will be proven 
that the mid-points F, E, D, of 
the three diagonals OC, A'B’, 
AB, lie on a straight line. 
Choose the sides OA and OB 
x as co-ordinate axes; represent 
Fig. 51. by a and a! the abscissas of the 
points A and A’; by b and Db! 
the ordinates of the points B and B’. The point D, middle of AB, has 


the co-ordinates 2! => y! =2. The point Z, middle of .A’B', has the 











; ee 
co-ordinates a/!=—, y!/=—. 
ae 2 


In order to get the co-ordinates of the point F, the middle of AC, seek 
those of the point C, which is the intersection of the lines AB’, A'B, 
whose equations are 

La. dae dete ed 
ae ee 
By solving these equations, the co-ordinates of the point C are found 
to be 
I See ! pg) 6 
op — au (6 b') yo (a—a!'). 
ab — a'b! ab — a'b! 





CHAP. I. STRAIGHT LINE AND CIRCLE. 81 


The point F being the middle of the line OC, its co-ordinates 2/!’, y’" 
are the halves of those of the point C’; one has, therefore, 


gill! — aa! (b — D') 


= _ bb' (a—a'). 
2 (ab — a'b!) 


~ 2 (ab — a/b!) 





m 
y 





Having the co-ordinates of the points D, EZ, F one can easily show that 
they lie on a straight line. The lines DE and DF have the following 
angular coefficients : 








bb! (a — a') 
yt—y b-bd oyM—y' ab—ab' si, 
at —g af —a wlll ad(b—b!) al —a’ 
ab—albt 


these two angular coefficients being equal to each other, it follows that 
the points D, EZ, F lie on a straight line. 


74. Prosiem VIII.— Find the angle between two lines. 
Let y= aa+b, y=a'x +b’, be the 








equations of two given lines. Draw ; ie 
through the origin, and on the same 

side of the axis as OY, two lines OA | Oe 
and OA! parallel to the given lines ier 
(Fig. 52); call wand a«'the angles which 7 ns 
they form with OX, V the angle which ee 


they inclose, and, to be definite, let 
a'>a. Evidently one has V= «'— a, whence 


! 
15 Ap ge tan a — tana. 
“ 1+ tane tan «! 





When the axes are rectangular, one knows that 
tana=a, tane’=a', 


if those values be substituted in the preceding formula, 


! 
(16 t ee 
(16) an V Gal 


82 PLANE GEOMETRY. BOOK II. 


In case the axes are oblique, one has (§ 61): 








a sin 6 a' sin 6 
tan a'= 


tan « = ‘ : 
1+ acosé6 1+ a'cos@ 


and, hence, 
17 AS a (a' — a) sin8 
Ch) mae 1+ aa'+ (a+ a’) cos 6 


One can deduce from these formulas the relation which must 
exist between the angular coefficients of two lines which are 
perpendicular to each other. In fact, in case the angle V is 
right, its tangent becomes infinite; one has, if the axes are 
rectangular, 


(18) 1 +aa'=0, 





and, if they are oblique, 
(19) 1+ aa'+ (a+a') cosd=0. 


75. Prosiem IX.— From a given point draw a perpendicular 
to a given line, and find the length of this perpendicular. 


2 ¢ 


Let (2) y=ax+b 


be the equation of the given line AB, a! 
and y' the co-ordinates of the given point 











a M (Fig. 53). Suppose the axes to be rec- 
: e * tangular. Any line passing through the 
Fig. 53. point MW has an equation of the form (§ 64) 


y—y=a' (a@—2'). 


In order that this line be perpendicular to the line AB, it is 
necessary that the relation 1+ aa'=0, be satisfied (§ 74) ; 


whence it follows that a! = ca, On replacing a! by its value, 
a 


one gets the equation of the perpendicular MP 


(20) y—-y'= — : (x — a), 


The co-ordinates x and y of the foot P of the perpendicular, or 
the point of intersection of the two lines AB and MP, are 


CHAP. I. STRAIGHT LINE AND CIRCLE. 83 


found by solving the simultaneous equations (2) and (20); but 
it is necessary to calculate the differences #— a! and y — y/' 
in terms of quantities which do not contain a and y (§ 55). 
Equation (2) can be written in the form | 


y—y' = a(x — x) — (y' — ax' — b); 


if, in this equation, y — y'is replaced by its value derived from 

equation (20), one finds 

Oy = dy 0) 

1 + a? 

and hence, by virtue of equation (20), 

y'— av'—b 
1 + a? 

By applying the formula for the distance between two points 

(§ 55), one gets the length / of the perpendicular MP, 


2s— oa 





b) 





yy = — 








2/7 ahs 2 = 4 [fy = aa! — 0) + a?) 
V (ae — FP + (y — y') G+ a) 





whence (21) l=+7 bel 
ae eae 

The sign is so chosen that 7 will have a positive value. It 
is easy to see that the numerator is positive or negative, accord- 
ing as the point Mis situated on the opposite or origin side of 
the line AB. For, let N be the point where the line AB is 
intersected by a line drawn, from the point M parallel to the 
axis of y; the point N being on the line AB, the ordinate y, of 
this point will equal ax’ + b, so that the formula (21) becomes 





jeri | Gane 
“Vita 
The difference, y' — y,, is positive in the first case and negative 
in the second. | 
It is to be noticed that the length of the perpendicular under 
this last form may be obtained immediately, by noticing that 
the right-angled triangle MNP gives 


MP = MNsin MNP = + (y'— %) cos «@ 


we ee aa 
sec @ V/1 +a? 





84 PLANE GEOMETRY. BOOK II. 


76. Suppose that the axes are oblique; the lines AB and 
MP will be perpendicular if their angular coefficients a and 
a' satisfy the relation 1+ aa! + (a+ a')cos6=0 (by 17); 








whence a’ = — cdaGeo8 e Therefore the equation of the 
a+ cos 6 
perpendicular MP is 
1+ acosé 
22 y—-y=— — wa’). 
- oe a+ cos 6 ct 


By solving the two simultaneous equations (2) and (22), one 
gets the co-ordinates x and y of the point P. If, as above, one 
seeks the differences 2 — a’, y — y', one finds 


_ (y' — ax' — b) (a + cos 6) 
1+ a+ 2acos 6 
_ (y'—az' — b) (1+ 40088), 


— f= 
eee, 1 + a? + 2acos 0 : 


a2 — a! 





+) 





substituting these differences in the formula for the distance 
between two points (§ 56), 


l=4+V@—2'P+y—y'h +2 @—-a/) y —y') cos 8, 
one gets x 


V(a + cos @)2+ (1 + acos 0)? — 2(a + cos @)(1+a C08 8) cos 8 . 
= 1+ 2acosé+ a 








i= 





(y' — ax! — bd) 
By developing, one remarks that the quantity under the radical 
contains the factor 1 — cos’6 or sin’, and is equal to 
(1 + 2acos 6 + a’) sin’ 6; 
(y' — aw' — b) sin 6 
V1 +2acosé + a 
The numerator will be positive or negative according as the 


point M is situated on the side of AB opposite to, or on the 
same side as the origin. The sign is so chosen that / is positive. 
‘ 





hence (23) l=+ 








77. In what precedes, we have supposed that the equation 
of the given line has the form y= aae+b. If the equation has 
the general form 
(1) Ax + By + C=9, 


CHAP. I. STRAIGHT LINE AND CIRCLE. 85 


the angular coefficient a of the given line being equal to -4 


one will have, in case of rectangular co-ordinates, a’ = — ; ae 5, 
and the perpendicular let fall from M will be represented by 
the equation 





! 
or (24) Pie tela 


Formula (21), in which one substitutes for a and 6 their 


values — 4 — 4 becomes 





Ag'+ By'+C 
Vat B 


This formula is an expression for the distance of a point 
from a straight line in rectangular co-ordinates: the numerator 
is the first member of the equation of the line, in which a and 
y are replaced by the co-ordinates of the point; the denomina- 
tor is the square root of the sum of the squares of the coeffi- 
cients of w and y. 

When the axes are oblique, one has 


_ B-—Acosé, 
A — Boos6’ 


the equation of the perpendicular will be 


(25) lst 





a! 


(26) he ee ee 
A—Bcosd@ B— Acosé 


and formula (23) becomes 


(27) tee (Aw! + By! + C) sind — 
V A? + B? — 2 ABoos 6 


It is easy to determine the sign of the numerator, according 
to the position of the point M with respect to the line AB. 
Let N (Fig. 53) be.the point where the line AB is intersected 
by MQ drawn parallel to the axis of y; imagine a movable 
point, having w and y for co-ordinates, to travel along this 











86 PLANE GEOMETRY. BOOK Il. 


parallel, and consider the values of the polynomial Av+ By+C 
for the various positions of the movable point. If the movable 
point be at WN, the value of the polynomial is zero. If the 
coefficient B is positive when one travels in the direction of 
positive y’s, the term By increases, and the function takes 
greater and greater positive values; when one travels in the 
opposite direction, it takes negative values; the contrary is 
true when B is negative. 


78. ProspLem X.— Through the point of intersection-of two 
given lines, draw a line perpendicular to a given line. 
Let Ax + By +C =0, 
A's + Bly +C' =0, 
Ale + By + C"=0, 
be the equation of three lines in rectangular co-ordinates. 


Every line passing through the intersection of the first two is 
represented by an equation of the form 


(Az + By+ C)+2A(A'e + By + C) =0; 
in order that it be perpendicular to the third, one must have 
AN(A+AA4) _ og. 
BiB AB c! 





tI 
whence one finds 
AA" + BB" 
Alan + BIB" 
On replacing A by its value, one obtains the equation sought, 
(28) (A'A" + B'B") (Ax + By + C) 
= (A"A + B"B) (A'x + Bly + C’). 


A= 





79. The three given lines form a triangle whose vertices are 
the intersections of these lines two by two. Equation (28) 
represents the perpendicular let fall from one of the vertices 
to the side opposite. By permuting the accents, one gets the 
equation of the perpendiculars let fall from each of the other 
two vertices to its opposite side, #.e., 


(A" A+ B"B)(A'e + Bly + C')=(AA' + BB) (Alle +B"y +0"), 
(AA'+ BB! (A"e + B"y + C")=(A'A" + BIB") (Ax + By + C). 


CHAP. I. STRAIGHT LINE AND CIRCLE. 87 


By adding the first two of these equations member to mem- 
ber, one obtains the third. Hence one infers ($ 70) that the 
three altitudes of a triangle pass through a common point. 


80. Prostem XI.— To find the locus of all points equally distant 
from two given points. 

Suppose the axes to be rectangular, and let x and y’, x’! and y'’ be the 
co-ordinates of the two given points. If x and y are the co-ordinates of 
any point whatever of the locus, the equation of the locus will be 


Zev te 9) -O-2 Fre), 
or, more simply, 


9) (a — a(n EEF) 4 yn yy(y FEW) <0, 


This locus is a straight line perpendicular to the line joining the two 
given points at its mid-point. 


81. Prostem XII.— To find the locus of all points which are 
equally distant from two given lines. 
Let us suppose the axes to be rectangular. Let 


Ax + By +C =0, 
A'v + Bly + C'=0, 


be the equations of the two given lines. If one represent the 
co-ordinates of any point of the locus by x and y, the equation 
of the locus will be 

(30) Aa + By + C_,Ae + Bly +C' 


Vat B VA" + B® 








Owing to the double sign, this equation represents two lines, 
which are the bisectors of the angles which are formed by the 
given lines. 


88 PLANE GEOMETRY. BOOK II. 


EQUATION OF THE STRAIGHT LINE IN POLAR 
CO-ORDINATES. 


82. Let O be the pole and OX the polar axis. The position 
of a line AB can be determined by the 
length a of the perpendicular let fall from 
the origin on this line, and by the angle 
« which this perpendicular makes with 

: A the polar axis, this angle having the limits 
r ax * Oand 2m. Let p and w be the co-ordinates 

Fig. 54. of any point of this line; by projecting 
the radius vector OM on the perpendicular OD, one has 








a . 
COS (w — a) 





(31) pcos (w—a)=a; or p= 


Since a and «@ are constants, this equation can be given the 
form, by developing cos (w — a), 


C 


2 = ' 
(32) “s Acosw+ Bsinw 





Conversely, every equation of this form represents a 
straight line; for, by referring it to rectangular co-ordinates, 
i.é., by taking the polar axis as the a-axis, and a perpendicu- 
lar to it at the pole as the y-axis, then using the transforma- 
tion formulas *=pcosw,y=psinw, the new equation is 
Ag + By C. . 


Remark. —If the line pass through the pole, then a = 0 and 
p, in equation (31), is not zero; therefore cos (w — a) = 0, or 


3 
o=at%, at 3y,.., 


Le. w = constant. 


CHAP. I. STRAIGHT LINE AND CIRCLE. 89 


ANOTHER FORM OF THE EQUATION TO A 
STRAIGHT LINE. 


83. Equation (31) developed becomes: 
p COS w COS @ + p SiN w Sin a = G, 
or, in rectangular co-ordinates, 
(33) acosa+ysina—a= 0. 


The equation of the line being put under this form, its first 
member has a very simple geometric meaning. Let any point 
M of the plane, whose polar co-ordinates are p and w, and rec- 
. tangular co-ordinates x and y, be consid- y; 
ered; from this point drop a perpendicu- a 
lar MP on the line AB (Fig. 55). The M 
projection of the radius vector OM on a 
the line OD is p cos (o—«); but this 
projection is equal to OD, increased or Pa m 
diminished by the perpendicular PM, A 
according as the point M and the origin ~ 
O are situated on opposite sides or on 
the same side of the line; if therefore this perpendicular be 
represented by p, affected with the + sign in the first case, 
and by the — sign in the second, one will have, in general, 








Fig. 55. 


a+ p= pcos(w—«a)=xcosa+ysina, 
whence p=+(ecosa+ysin a — a). 


Thus, the first member of the equation (33) represents the 
distance from any point of the plane, whose co-ordinates are 
2 and y, to the line represented by this equation, this distance 
affected with the proper sign. 

It is easy to deduce the co-ordinates 2, and y, of the foot P 
of the perpendicular; the differences 7 — a, y — %, being the 
projections of the line PM on the two axes, we have 


e—% = pcosa= (xcosa+ysina — a)Ccosa, 


Y¥—Y=psina=(wcosae+ ysina —a)sina. 


90 PLANE GEOMETRY. BOOK It. 


The form (83), under which the equation of the line can 
always be put, is useful in a large number of investigations. 


83. 2. The equation of a line passing through two points. 


Let 
(pi, 1); (pe W») 


be the co-ordinates of the two points; the equation of the line 
joining these two points is 


cosSw sinw 
cosw, sine, | = 9; 


COS w, SIN ws 








Sei lp >Ie 


in fact, this equation represents a straight line since it can 
be put under the form of (32), and it is evidently verified by 
p= py © = and p = po w = wy. 


CHAP. IL. THE CIRCLE. 91 


CHAPTER II 
THE CIRCLE. 


84. We seek first the equation of the circumference of a 
circle in rectangular co-ordinates. Repre- ¥* 
sent by a and b the co-ordinates of the 
center C (Fig. 56), and by r the radius; 
the circumference, being the locus of the 
points whose distance from the center 
is equal to the radius, has for its equa- ~ ° x 
tion 


(1) (@—a)y+y—bf=r; 
this equation developed may be written 
(2) A(e’ + y’?) +2 De +2 Ey+ F=0. 


Hence, the equation of the circle, in rectangular co-ordinates, 
is an equation of the second degree, which does not involve the 
product xy of. the variables, and in which the terms in a and y 
have the same coefficient. 








Fig. 56. 





85. Conversely, every equation of this form, in rectangular 
co-ordinates, represents a circumference of a circle, if it repre- 
sents a locus. In fact, equation (2) can be written in the form 


; 2 2 2 2 





A Ae A 


The center C will, therefore, by (1), have the co-ordinates 


-= and —=; the first member represents the square of 


the distance of any point M of the plane, having the co- 
ordinates « and y, from the point C; if the second member 
is positive, the equation will be satisfied by the co-ordinates 


92 PLANE GEOMETRY. BOOK II. 


of all the points of the plane whose distance from C is equal | 








to vz +4 F, it represents therefore a circumference 
of a circle. When the second member is zero, the distance 
MC becomes zero, the point M will coincide with the point C, 
and the equation will still be satisfied by the co-ordinates of 
this point; the locus will be reduced therefore to a single 
point. 

Finally, in case the second member is negative, the equation 
cannot be satisfied by any point of the plane; because the 
square of the distance of the point M from the point C is a 
positive quantity; the equation cannot therefore, in this case, 
represent a geometrical locus. 


86. Suppose now that the co-ordi- 
nate axes are oblique, and inclose an 
angle 8 (Fig. 57); by expressing that 
the distance of any point of the locus 
from the center is equal to the radius, 
one will have the equation of the 
Fig. 57. circumference, 


(3) (x — a)? + (y — b)? + 2 (x — a) (y — b) cos =". 
This equation may be written in the form 


(4) A(x’? + y +2 xy cos 6) +2 Da+2 Ey+ F=0. 








Hence, the equation of a circle, in oblique co-ordingtes, is an 
equation of the second degree, in which the terms in x’, in y’, and 
in 2 xy cos have the same coefficients. 

On dividing by A, one reduces, as in (3), the coefficients of 
x, y’, 2 xy cos @ to unity. 


87. Conversely, every equation of this form represents a 
circumference of a circle, if it represent a locus. In fact, one 
can determine the three constants, a, b, and 7°, by comparing 
equations (3) and (4). Equation (3), developed, becomes 


a + y’ + 2 xy cos 6 — 2 (a + beos 0)a — 2 (b + acos O)y 
+ @?+ 8°42 abcos§—7?=0. 


rare nm. 2. THE CIRCLE. 93 
Equations (3) and (4) will become identical by placing 


a+b 0030 =—= b+ acos 6 =—=, 


at 4 Pabeos 6 P= 
A 
The first two relations give finite values for a and 6, since 
the determinant 1— cos’6 or sin’@ is different from zero. 
The third gives 
yi ght B42 ab cos 8 — 
A 
Notice the point C, which has the co-ordinates a and b. The 
first member of equation (3) represents the square of the 
distance of any point M of the plane, whose co-ordinates are 
aw and y, from the point C. If 7° is found to be a positive 
quantity, the equation will be satisfied for every point of 
the plane whose distance from C is equal to r; it represents 
therefore the circumference of a circle. If 7? has the value 
zero, then the distance MC equals zero, and the equation will 
be satisfied by the co-ordinates of the point C; it will repre- 
sent a single point. Finally, if 7? have a negative value, the 
equation will not be satisfied by a single point of the plane. 
Instead of determining the center C of the circle by its 
co-ordinates a and b, it is more convenient to determine it by 
the orthogonal projections of the line OC on the two axes. 
Call these two projections OD and OE, a' and b' (Fig. 57), 
affected by proper signs, and express the fact that the projec- 
tion of the line OC on the one or on the other axis is equal to 
that of the broken line OPC or OQC; one has 


a'=a+bcos6, b'=b+acos 6; 


whence, a! = ~4. Me -= After having laid off the 


lengths a’ and 0b! on the axes, beginning at the origin, one 
erects perpendiculars to the axes at the points D and E; the 
intersection of the two perpendiculars will determine the 
center C. 


9 PLANE GEOMETRY.  — BOOK II. 


88. The equation of the circumference of a circle, as has 
been found, is 


(5) (@—a)?+ GOO ee Gy pie 0. 


The first member has a geometrical signification which it is 
well to notice. Consider a point M of the plane having the 
co-ordinates « and y; the expression 


(a — a)? + (y — b)? + 2 (@ — a) (y — b) cos 6 


represents the square of the line MC, which joins the point 
M with the center (Fig. 58); the first member of the equation 
is, therefore, equal to MC? — 7°, that is, to the product of the 

two factors MC +r and MC —r, which are 
* the two segments WA and MB of the diam- 

eter drawn through the point M, the seg- 

ments being affected by the same or contrary 
. signs, according as they are measured in 
the same or in opposite directions. Thus the 
first member of equation (5) represents the 
product of two segments of any secant drawn from the point 
M, that is, the power of this point with respect to the circle. 
When the point M is without the circumference, this product 
is equal to the square of the tangent. 





Fig. 58. 


89. Prosiem I.— To find the equation of the tangent to any 
curve. . 

We have already given the definition of a tangent at a point 
M of a curve (§ 19). Through 
the point M and a neighboring 
point M' on the curve draw a se- 
cant MM", and allow the point M' 
to continually approach the point 
M. The secant MM' will revolve 
about the point M, and if it tends 
toward a limiting position MT, 

Fig. 59. this line MT is called the tan- 
gent to the curve at the point M (Fig. 59). 











' 
' 
' 
1 
' 
1 
' 
, i 
Oo | P 


CHAP. II. THE CIRCLE. 95 


Let x and y be the co-ordinates of the point of contact M; 
x+thand y+k those of the neighboring point M'; the angu- 


lar coefficient of the secant MM' is the ratio ; of the difference 
r} 


of the ordinates of the two points Mand M' to the difference 
of their abscissas. As the point M’ approaches indefinitely 
toward the point M, the two increments h and k tend simulta- 
neously toward zero; we study here curves defined by equa- 


tions such that the ratio ; tends toward a limit, which is the 
’ 


derivative of the ordinate regarded as a function of its abscissa. 

If the equation of the curve is solved with respect to y, and 
put under the form y=/f(«), the tangent will have for its 
angular coefficient y!=f'(x). In case the equation of the 
curve f(a, y)=0 cannot be solved, the derivative y' of 
the implicit function y can be derived from the equation 
fi.+y'-f',= 9, in which f', and /', represent the partial 
derivatives of the function f(a, y), with respect to a and y. 
Whence it follows 


(6) eee 


Thus, if X and Y be the co-ordinates of any point of the tan- 
gent, the equation of this line will be 





! 
(1) Y-y=— F(X — a), or (XO + (VF 4=0. 

y 

90. Prosiem II. — To find the equation of the tangent to the 
circle. 

Let the preceding formula be applied to the circle, suppos- 
ing that the axes are rectangular and the origin is at the center 
of the circle. The equation of the circle is 


(8) + y—r=0. 


The equation, solved for y, becomes y= -+ V7"? — 2’; on tak- 
ing the derivative of this function, one has 





96 PLANE GEOMETRY. BOOK It. 


By leaving the equation unsolved and applying formula (6), 
the same value y' = —~ is obtained. Thus the equation of the 
tangent is : 


¥—y=—"(X-2), or aX +yY=H=e?4 y7 


Since the point M is on the circle, its co-ordinates satisfy 
the equation of the circle, and one has a? + y? = 7°. equa- 
tion of the tangent is simplified and becomes 


(9) GX + yY <2". 
Since the angular coefficient of the radius drawn to the 


point of contact is Y, it follows that the tangent is perpendicu- 
lar to this radius. 


91. Prosiem III.— To draw a tangent to a circle from an 
exterior point. 

Suppose that the circle is always referred to rectangular 
axes drawn through the center, and represented by the equa- 
tion . 


(8) w+ par; 


call a, and y, the co-ordinates of the given point P (Fig. 60). 
Let MP be a tangent drawn from this point; the question is 
now to determine the point M, whose unknown co-ordinates 
are assumed to be a and y..: The point 
M being on the circle, its co-ordinates 
satisfy equation (8). The tangent at 
the point M has the equation #X + y¥Y 
=7*, This tangent passes through the 
point P, and the equation must be satis- 
fied by the co-ordinates of this point, 
which furnishes the relation 


(10) LX + YY, = .. 











Fig. 60. 


By solving the two simultaneous equations (8) and (10), the 
values of the unknown 2 and y will be obtained. 


CHAP. II. THE CIRCLE. 97 


The solving of the two equations (8) and (10) results in 
finding the point of intersection of the two lines. The first 
equation represents the given circle; the second a straight 
line. To find the values of x and y, which satisfy at the same 
time these two equations, is to find the points of intersection 
of the line and circle. This line cuts the circle in two points 
M and M' and is the line of contact. It is to be noticed that 
equation (10) of the line of contact has the same form as 
equation (9) of the tangent; only, that the co-ordinates of the 
point of contact are replaced by those of point P. 


92. One knows that, in case one has two equations 
A eae 


simultaneous with respect to two unknown quantities # and y, 
if one of the equations be replaced by mA + nB = 0, which is 
obtained by multiplying the equations by the arbitrary quan- 
tities m and n, and then adding them member to member, one 
forms a new system of equations 


A=0, Am + Bn= 0, 


equivalent to the given system. This signifies geometrically 
that the points of intersection of the two curves represented 
by the two given equations are the same as the points of inter- 
section of one of them with the third curve. 

It has been stated that the points of contact M and M ' are 
given by the intersection of the given circle and the line of 
contacts. By subtracting the two equations (8) and (10) mem- 
ber from member, one obtains the new equation 


e+ yy — ae — ny =0, 


ee WY _ oP + ye 
or Ser aon —_— — = 
(°-3) +(0- 9) <a 
which may replace equation (10). This new equation repre- 
sents a circle whose center is the mid-point of the line OP and 


has the co-ordinates 3 and o Since the equation has no con- 


stant term, the circle passes through the origin and is, there- 
G 


98 PLANE GEOMETRY. BOOK It. 


fore, described on OP as a diameter. The points in which the 
circle cuts the given circle are the points of contact. In this 
manner the construction of elementary geometry is verified. 


93. Prositem IV. — To draw a tangent parcilel to a given line. 
To the circle 


(8) gi y = a 


it is required to draw a tangent parallel to the line OA, which 
is supposed to be drawn through the origin, and to be repre- 
y sented by the equation y = ma (Fig. 
Es 61). If wand y be the co-ordinates 
of the point of contact M, one knows 
that the angular coefficient of the 





> NF tangent is equal to ae In order 


that the tangent MT be parallel to 
the given line, the angular coefficient 





. wv 
must be equal to m, te. —— =m, or 
y 


1 

(11) am 

Further, the co-ordinates of the point M satisfy the equation 
of the circle. These co-ordinates are therefore determined ‘by 
the two simultaneous equations (9) and (11), and, consequently, 
the points of contact M and M' are given by the points of 
intersection of the circle and the line represented by equation 
(11). It may easily be shown that the line MM’ is Od oc 
dicular to the line OA. 


94. This problem may be discussed in another manner, and 
this will give us the opportunity of presenting the equation of 
the tangent to the circle in a new form. Let us therefore seek 
the points of intersection of the circle a + y’ = 7°, by the line 
y=mx-+k. On eliminating y, one gets the equation of the 
second degree, x? + (ma +k)? = 7°, or 


(m? + 1)2?+2mke+h?-r?=0. 


CHAP. II. THE CIRCLE. 99 


When this equation has real roots, the line cuts the circle in 
two real points whose abscissas are the roots of the equation. 
In case the roots of the equation are equal, the points of inter- 
section will coincide, and the line becomes a tangent to the 
circle. Finally, when the roots are imaginary, the line does 
not cut the circle. 

Thus, the condition that the line be tangent to the circle is 


mk? = (m? + 1) (kh? — 2°), or RP =r? (m? +1). 
Substituting for k its value, the equation of the line becomes 
(12) y= me trVm? +1. | 


This equation, which involves a single arbitrary parameter 
m, represents all the tangents to the circle. 


95. Propiem V.— To Jind the locus of the points whose distances from 
two jixed points are in a given ratio. 

Let A and B be the two given points y 
(Fig. 62). Take the line AB as the 
x-axis, and for the y-axis a perpendicular 
to AB at its middle point. If one calls 


2 a the distance AB, cr 


™ the given ratio, 

n 
and if x and y designate the co-ordinates == 
of any point in the locus, the equation 4 es Oe ae B 
of this locus will be 











y? +(* +a)? _ m? : 
ye +(e — ay? a 7’ F Fig. 62. 
m2 
or (13) a2 4 y2?—2axMt™ 4 a =0. 
m2 — n2 


This is a circle whose center lies on the axis of XY. The two extremities 
of the diameter DE are the points which divide the line AB in the ratio 
m to n. 


96. Prosiem VI. — To find the points of intersection of two 
circles. 

Let (14) w@+y¥+2De42Hy + Ff =), 
(15) et+yt2D'e+2 E'y4+ F'=0, 
be the equations of two circles in rectangular co-ordinates, the 
coefficients of 2°+y? being equal to unity. The points of 


100 PLANE GEOMETRY. BOOK II. 


intersection will be given by these two simultaneous equations. 
One can replace the second circle by the line 


(16) 2(D— D)a2#+2(E— E)y+(F-F)=0, 


which is obtained by subtracting the equations member from 
member, and the question is reduced to finding the points of 
intersection of the first circle with this line. If the line cuts 
the circle, the circles have two points of intersection, and 
equation (16) represents the common secant. If the line 
becomes tangent to the circle, the points of intersection coin- 
cide, and the two circles will be tangent; equation (16) will in 
this case represent the common tangent. Finally, when the 
line does not cut the circle, the two circles do not have a 
common point. 

Moreover, the equation (16) has in every case a remarkable 
geometrical signification. The first members of equations (14) 
and (15) represent respectively ($ 88) the powers of any 
point M of the plane, having the co-ordinates a and y, with 
respect to the two given circles; whence equation (16) may be 
obtained by equating these two expressions, the terms of the 
second degree canceling; equation (16) represents, therefore, 
the locus of points of equal power with respect to two 
circles; this locus is a straight line, which is called the radical 
axis of the two circles. The portion of this line external to 
the circles is the locus of the points from which any pair of 
tangents drawn to the two circles are equal each to each. It 
is clear that the radical axes of three circles meet in a point; 
this point is called the radical center of the three circles. 
When it is exterior to the three circles, the tangents emanating 
from this point have the same lengths. The circle described 
about this point as center with a radius equal to the common 
length of the tangents is orthogonal to the three circles con- 
sidered. 


Remarx.—If the coefficients of a?+y? be not equal to 
unity, and if the equation of the two circles are of the form 


(17) S(@, y =A (ww +y7)+2 De +2 Ey + F=0, 
op (x, y =A (2+ y)4+2De+2Ey+ F'=0, 


CHAP. II. THE CIRCLE. | 101 


the equation of the radical axis can be derived by eliminating 
the terms of the second degree between the two equations, that 
is to say, by multiplying the first by — A'the second by 4, 
and adding; thus is found the equation 


(18) Ap — Alf=2 (AD! — DA) a+ 2(AE'—EAy 
+ AF' TA <0, 


More clearly does this equation represent the radical axis, 
because the power of the point (a, y) with respect to the first 


circle 1s Pew , with respect to the second, Pe, by equating 


these two powers and clearing of fractions, one gets equa- 
tion (18). 


97. Prostem VII.— To find the general equation of the 
circles which pass through the points of intersection of two given 
circles. 

The totality of these circles is called a pencil of circles. 
Their equation may be found by a method identical with that 
employed in the analogous problem of straight lines (§ 69). 
Let the two circles be represented by equations (17), the 
equation 


(19) S(@, y) + rA> (a, y) = 0, 
that is to say, | 
(A +2A (a? + y?) + 20D+ AD'\x +2(E +2XE")y +PreAr =o 


where A plays the réle of an arbitrary constant, represents a 
circle passing through the points of intersection of the two 
given circles; for the co-ordinates of each of the points of inter- 
section reducing f and ¢ to zero, evidently make f+ Ad = 0. 
Equation (19) is the most general equation of the circle sought, 
that is to say, for every value of A, it represents some circle S 
passing through the points common to the given circles. In 
fact, choose a point (a, y) on the circle S, and determine » by 
the equation of the first degree 


S(@y %*) + Ap(*r; 1) = LIBRARyS, 


UNIVERSITY } 










OF CALIFORWK, 


102 PLANE GEOMETRY. BOOK It. 


which expresses the condition that the circle (19) passes 
through the point (a, y). The coefficient A being thus de- 
termined, the circle (19) and the circle S coincide, because 
they have in common three points at finite distances, namely, 
the two points of intersection of the given circles and the given 
point (a, 4). 

All of the circles of the pencil (19), taken two by two, have 
the same radical axis, which is no other than the radical axis 
of the given circles (18). This radical axis is to be reckoned 
among the circles of the pencil, for it is derived by giving the 


particular value -= to , which causes the terms of the 


second degree to disappear. 


Limitine Points. — Take, for simplicity, the line through 
the centers of the given circles as the z-axis and their radical 
axis as the y-axis; the equations of the two circles take the 
form 


20) e+y—2ax+c=0, 
e+y—2a'e+c=0, 
a and a! being the abscissas of the centers of the two circles 
and ¢ the power of the origin with respect to each of the two 
circles, the power being the same for the two circles, because 
by hypothesis the origin is on the radical axis. The general 
equation of the circles passing through the points common to 
the two circles is : 
(1 +A)(@’+ y’) —2(a+da'a+(1 + A)e = 0, 
or more simply on dividing by (1+A) and calling the ratio 





a+ da’ _ 
ares ae we 
(21) e+ y?—2yr+c=0, 


where yp represents an arbitrary coefficient. This last equa- 
tion could have been deduced a priori, because it is the general 
equation of the circles, which, associated with either of the 
given circles, has OY as the radical axis. 


CHAP. II. THE CIRCLE. 103 


Among the circles (21), there are two, each of which reduces 
to a point real or imaginary, or, in other words, has a radius 
equal to zero; these two circles are the limiting points. Equa- 
tion (21) can be written 


(w@— py +y=pi—e. 


Therefore, for »=+Vé¢, this circle reduces to the point 
a=p,y=0. If cis positive, that is, if the origin is eaterior 
to the two circles, or what amounts to the same thing, if the 
two circles intersect in imaginary points, the values of p are 
real and the two limiting points are real. In this case, Ve 
represents the common length of the two tangents drawn from 
the point O to the given circles; the limiting points are there- 
fore the intersections of the line of centers, Ox, with the circle 
described about the foot O of the radical axis as center, with a 
radius equal to the length of the tangent drawn from O to any 
one of the given circles. If, on the other hand, the two given 
circles (20) intersect in real points, O being within the two 
circles, ¢ is negative and the two limiting points are imaginary. 
If the two given circles are tangent, O is their point of contact, 
c= 0, » = 0, and the two limiting points coincide with O 


97. 2. Proptem VII.-— Find the condition that two circles 
intersect orthogonally. 


When two circles intersect at right angles, the radii drawn 
to the point of intersection M are perpendicular, because they 
are perpendicular to tangents which are perpendicular by 
hypothesis. The triangle, which has as vertices the point M 
and the two centers, is therefore right angled at M, and the 
square of the distance between the centers is equal to the sum 
of the squares of the radii. Suppose that the two circles are 
represented by equation (17); then by using the expressions 
given in § 85 for the co-ordinates of the center of a circle 
and the square of its radius, the condition that the two circles 
cut orthogonally in rectangular co-ordinates will be 


(22) AF! + FA! —2(DD' + EE') =0. 


104 PLANE GEOMETRY. BOOK It. 


The same result can be derived without the assistance of 
geometry. Let (2, y) be a point common to the two circles (17): 
the angular coefficients of the tangents to the two circles at 


this point being respectively ($ 89) — s and — a the neces- 
y y 

sary and sufficient condition that the two circles be orthogonal 

at the point (a, 7) is 


tL $', +f", $', ae 0, 
which becomes by substituting and developing 


(23) AA'(a? + 9°) + (AD! + DA)a + (AE' + A'E)y 
+ DD' + EE' =0. 


If x, y be regarded as current co-ordinates, this last equation 
represents a circle, and, as it should be satisfied by the points 
of intersection of the two given circles (17), it ought to repre- 
sent a circle passing through the points of intersection of these 
two circles. Moreover, the three circles (17) and (23), taken 
two by two, should have the same radical axis. The radical 
axis of circle (23) and the first circle, f= 0, of (17) has the 
equation | 


(AD'— DAx + (AE'— EA)y + DD! + EE'— FA'=0; 


the equation of the radical axis of circle (23) and the second 
circle ¢ = 0, of (17), is vf 


(AD'— DA)zx + (AE'— EA')y — DD! — EE'+ AF'=0. 


Expressing the condition that these two equations should 
be identical, one gets equation (22). 

The condition expressed in (22) may be verified by suppos-" 
ing A'=0; the second circle becomes a straight line, and 
the condition of orthogonality ought to express the condition 
that this straight line pass through the center of the first 
circle. 


Remark.—The condition of orthogonality is linear and 
homogeneous with respect to the coefficients of each of the 


CHAP. II. THE CIRCLE. 105 


circles. Conversely, if between the coefficients A, D, LE, F 
of the equation of a circle 


A(a? + y*)+ 2 Dx +2 Ey + F=0, 
any linear-and homogeneous equation 
LA+MD+NE+PF=0 


be established, this relation compared with condition (22) 
shows that the circle considered is orthogonal to the fixed 
circle 

Pia? + y’)— Ma — Ny+ L=0. 


Apprications. —To find the equation of a circle which cuts orthogo- 
nally three given circles 


f(t, y) =A? +y?)4+2De 4+2ky +F =), 

o(a, Y= Ale? + y2)4+2D'e +2by + F =), 

W(x, y)= A" (a2 + y2) + 2 Dla + 2 E"y + FY =0 
Let. (24) a(x? +°y2)4+ 2dx + 2ey+ f=0 


be the circle sought; one should have, after condition (22) has been 
applied to circle (24), associated with each of the other three in order 
and been arranged with respect to a, d, e, and f. oa 


(25) aF —2dD —2eE +fA =0, 
aF! —2dD! —2eE' +fA' =0, 
aF" —2dD" —2eE" + fA" =0. 


If the three given circles taken two by two do not have the same radical 
axis, these equations give a single system of values for the ratio of any 
one of the coefficients a, d, e, f to the other three; there is, therefore, 
one circle only cutting the proposed circles at right angles; it is called 
the orthotomic circle. Its equation is obtained by eliminating a, 2d, 2e, 
f between equations (24) and (25), which gives in determinant form 


xe? + y?, mes ee 5 1 
F, D; ££, A 
F, Dow A! 
F'"', D", Dae All 


= 0. 


It is admissible to suppose that any one of the coefficients A, A!, A’! is 
zero ; the corresponding circle is then replaced by a straight line. 


106 PLANE GEOMETRY. BOOK It. 


Net or Crrcues. — Let f(2, y), > (2, y), ¥ (a, y), be the first 
members of the equation of the three preceding circles, which, 
taken two by two, do not have the same radical axis; the 
equation 


(26) Af (x, Y) + pp (x, Y) + wy (a, y) = 0, 


where A, p, v are arbitrary coefficients, represents an infinitude 
of circles, forming what is called a net. It is desired to 
determine the condition that every circle of the net be 
orthogonal to some fixed circle, that is, orthotomic to a circle. 
In fact, by adding equations (25), member to member, after 
having multiplied the first by A, the second by p, the third by 
v, a relation is obtained which expresses precisely that the 
circle (26) is orthogonal to the orthotomic circle (24). 

Conversely, the totality of the circles which are orthogonal 
to a fixed circle forms a net. For the condition that a 
circle S be orthogonal to a fixed circle leads to a linear homo- 
geneous relation with respect to the four coefficients of the 
equation of the circle S. One of these coefficients is therefore 
a linear homogeneous function of the other three, which are 
arbitrary and which may be called A, », v; the equation of the 
circle S arranged with respect to A, pw, v takes then the form 
of (26), and the circle S forms a net. 


EQUATION OF A CIRCLE IN POLAR CO-ORDINATES. 


97. 3. Let O be the pole and OX the polar axis (Fig. 63) ; 
call a and « the co-ordinates of the center C, r the radius, 
and p and w the co-ordinates of any 
= point M of the circumference. In the 

triangle OCM, one has 


(27) p’ — 2ap cos (w — a) +H— r= 0. 


When the pole O is situated on the 
* circumference, one has a= 7, and the 
equation reduces to 


(28) p = 27 cos (w — @). 





Fig. 63. 


CHAP. [l. THE CIRCLE. 107 


As an application of this equation, consider'two circles which 
intersect; through one of the points of intersection O, draw 
any secant; this secant meets the circles in two other points 
Mand M'; find the locus of the mid-point of the line MM’. 
If the point O be taken as pole, the two circles are represented 
by the equation 


p = 2rcos(w— a), p=27r'cos (w— «'), 
and one obtains immediately the equation of the locus 
p=7 Cos (w— @) + 1! Cos (w — @'); 
this equation can be put under the form 
p= 27, COs (w— &), 


and the locus is a circle passing through the point of intersec- 
tion O of the two given circles. 


108 PLANE GEOMETRY. BOOK II. 


CHAP Ei it 
GEOMETRICAL LOCI. 


98. Geometrical loci may be defined in various ways. 
Whenever a property common to all points of a locus is given, 
it is by interpreting this property by means of algebraic 
symbols that the equation of the locus is obtained. In this 
manner, the circumference of a circle was defined as the locus 
of points whose distances from a given point are equal to the 
same given quantity; it was by expressing this property, 
common to every point of the locus, that the equation of the 
circumference was obtained (§ 84). Thus, also, has been 
found the locus of the points whose distances from two given 
points are in a given ratio ($ 95); the expressing of this 
property gives the equation of the locus. Likewise, by the 
same process, the equation of the perpendiculars erected at 
the mid-point of the straight line which joins two given points 
(§ 80), and those of the bisectors of the angles formed by 
two given lines (§ 81). 

But, usually, a curve PQ (Tig. 64) is defined by ‘the motion 
of a point in the plane. Each position of the variable point 
M is determined by the construction of a figure whose various 
parts depend on an arbitrary parameter a. Consequently the 
two co-ordinates w and y of the point M are functions of this 
variable parameter a: let 


xv 1d), y = f(a), 


be the two functions; one sees that the equation of the locus 
described by the point W is found by eliminating the param- 
eter a between the two equations. 

More generally, the geometric construction determines every 


CHAP. III. GEOMETRICAL LOCI. 109 


point of the locus by the intersection of the variable curves 
which depend on the parameter a; let 


(1) F(a, Y, a) = 9, 
(2) : F(a, Y; a) oa 0, 


be the equations of the two curves. Ifa particular value be 
assigned to this parameter, two curves A and B are obtained 
which intersect in a point M, whose 
co-ordinates « and y satisfy the two 
simultaneous equations (1) and (2). 
If another value a! be assigned to the 
parameter, the two lines will occupy 
the positions A' and B', and the point 
of intersection will be at M'; a third 
value uw!’ assigned to the parameter 
will give the two curves A" and B" 
and their point of intersection M", and so on. Allow the 
parameter a to vary in a continuous manner; then the two 
curves A and B will be displaced in the plane in a continuous 
manner, and the point of intersection M will describe the 
line PQ. 

The equation of the curve PQ, the locus of the point M, 
will be found by eliminating the parameter a between the two 
equations (1) and (2). In fact, the elimination of a between 
the two equations (1) and (2) amounts to finding a system of 
two equations 


(3) F(a, Y; a) = 0, 
(4) S(@, y) = 0, 


equivalent to the system of two equations (1) and (2), and 
such that one of them does not contain the symbol a. Two 
systems of equations are said to be equivalent, when they 
are satisfied by the same assigned values of the variables. 
When a particular value is assigned to a, the co-ordinates x 
and y of the point M associated with this value of a form 
a system of three quantities, x, ¥, 4, which satisfy at the 
same time the two equations (1) and (2); since the system 





Fig. 64. 


110 PLANE GEOMETRY. BOOK II. 


of equations (3) and (4) is equivalent to the preceding, these 
values satisfy also equations (8) and (4); equation (4), which 
does not involve a, is therefore satisfied by the co-ordinates 
of every point of the locus. 

Conversely, every point M, whose co-ordinates # and y 
satisfy equation (4) belongs to the locus. Because, if one 
determines. a value a which satisfies equation (3), in which 
one gives to # and y the preceding values, one gets a system 
of values of three quantities, 2, y, a, satisfying the system of 
equations (3) and (4). Equations (1) and (2), constituting 
a system equivalent to this system, will be satisfied by the 
same values; one will thus obtain two lines A and B passing 
through the point . 

It can happen, moreover, that to a system of real values 
of w and y satisfying equation (4) corresponds a value of a, 
for which ‘equations (1) and (2) do not represent real curves; 
one will have this kind of a locus, for example, if the value 
of a were imaginary. But, in every case, the values of a, y, 
a will satisfy the two equations (1) and (2). 


99. Although the construction of each of the positions of 
the figure, which gives the various points of the locus, de- 
pends upon the value given to the arbitrary parameter, it is 
frequently more convenient to introduce into the discussion 
several variable parameters a, 0, ¢, ---; but these parameters 
are then so connected with one another that the value of one 
only is arbitrary, and that the variation of this’ parameter 
determines moreover the value of the others. If these param- 
eters are n in number, they will be connected by n — 1 equa- 
tions of condition. 

Suppose, for example, that only two variable parameters 
a and b connected by the equation of condition 


6) o(a, b)=0 
are employed, and let 
(6) : F(a, y, a, b)= 0, 


(7) F, (a, y, a, b)= 0, 


CHAP. III. GEOMETRICAL LOCI. 111 


be the equations of two variable curves A and B, whose inter- 
sections furnish every point of the locus. If the parameter 
@ varies in a continuous manner, the parameter 6 which de- 
pends upon a by reason of the relation (5) will vary also in 
a continuous manner; the two curves A and B, whose equa- 
tions contain the two parameters, vary also in a continuous 
manner, and their point of intersection M will describe a 
eurve PQ. 

The equation of this curve will be obtained by eliminating 
the two parameters a and } between the three equations (5), 
(6), (7). In fact, to eliminate a and 6 between these three 
equations is to find a system of three equations 


(8) F,(2, Y, a, b)= 0, 
(9) F(a, Y; a, b)= 0, 
(10) St (&, y= 0, 


equivalent to the given system, and such that one of them 
no longer contains a and 6. When values are assigned to 
a and b which satisfy equation (5), the co-ordinates x and y 
of the point M, associated with these values of a and 6, form 
a system of values of four quantities a, y, a, b, satisfying at 
the same time the three equations (5), (6), (7). The three 
equations (8), (9), (10) forming a system equivalent to the: 
preceding system will also be satisfied by the same values ; 
equation (10), being independent of a and b, will therefore 
be satisfied by the co-ordinates # and y of each point of the 
locus. 

Conversely, every point M whose co-ordinates # and y 
satisfy equation (10) belongs to the locus; because if one 
determines the values of a and } which satisfy the two equa- 
tions (8) and (9), in which # and y have been assigned the 
preceding values, one has a system of values of four quantities 
av, y, a, b satisfying the system of three equations (8), (9), (10). 
The three equations (5), (6), (7), forming a system equivalent 
to the former, will also be satisfied, and one will have two 
curves A and B passing through the point ©. 


112 PLANE GEOMETRY. BOOK It. 


100. Suppose, in general, that one employs variable pa- 
rameters a, b,c, ...,h connected by n — 1 equations of condition, 





(a, -0, ¢, +++, h)= 9, 
(ut) I ee 
dn —1 (a, b,c, +++, h)=O, 
and let 
(12) F(x, y; sol: b, ¢, +++, h) =0, 
(13) F, (a, y, a, 6, 6, ++, h)= 0, 


be the equations of two variable curves A and B, whose inter- 
sections give every point of the locus. When the parameter a 
is allowed to vary, the remaining parameters vary simultane- 
ously, and the point M describes the locus. The equation of 
this locus is obtained by eliminating the n parameters between 
n +1 equations (11), (12), (18). 


101. It has been asserted that the construction of the 
figure depends upon a single arbitrary parameter a. If the 
figure should depend upon two arbitrary parameters a and }, 
the two co-ordinates 2 and y of the point M would be func- 
tions of these two parameters, 7.é., 


a= f(a, b), y=fi@ >). 


Such values could be assigned to these parameters that the 
point M might be made to coincide with any point of the 
plane, having the co-ordinates 2, and y, To accomplish this, 
it suffices to determine a and b by means of the equations 


= f(a, 6b) n=h (a, 5). 


The point M may describe the entire plane and not any defi- 
nite curve in the plane. 

One sees very clearly, then, why it is necessary, when n 
variable parameters are employed, that these » parameters be 
connected by n—1 distinct equations of condition; because, 


f 


CHAP. III. GEOMETRICAL LOCI. 113 


if these equations of condition could be reduced to a smaller 
number, two parameters at least would be arbitrary. 

It is possible that the two variable curves A and B inter- 
sect in several points; the preceding process gives the locus 
described by the totality of these points. — 


102. Remark. —It often happens that one of the two vari- 
able curves A and B, whose intersection furnishes a point M 
of the locus, passes through a fixed point P. In this case, the 
co-ordinates of this point P satisfy the equation found by elim- 
ination. In fact, suppose that the equations of the two curves 
contain n variable parameters connected by n —1 relations 
(§ 100); if the co-ordinates #, and y, of a fixed point P sat- 
isfy the equation of the line A, whatever be the values of the 
parameters, by replacing x and y in the equation of the curve 
B by a, and y,, one will get an equation, which, combined with 
the n —1 equations of condition between the parameters, will 
form a system of n equations which will determine the values 
of these parameters. This point P will, properly speaking, 
be foreign to the geometrical locus, if imaginary curves corre- 
spond to the values found. 

In this case, it frequently happens that the point P enters 
in the equation through a particular factor which ean be 
removed. After this factor has been suppressed, the equa- 
tion represents the geometrical locus itself. But often it is 
impossible to decompose the first’ member of the equation into 
two factors, and the point P must be considered as an isolated 
point connected with the curve. 

103. Prosiem I. Being given in the plane (Fig. 65) an angle AO, } 
and a fixed point P, draw through the point P the fixed secant PBA and 
the variable secant PDC; draw also 
the lines AD, BC; find the locus of their 
point of intersection M. 

Take the lines OX and OY as co- 
ordinate axes, and represent by x; and y1 
the co-ordinates of the point P. The 
fixed secant PBA will have an equation 
of the form 

y—yi=a(e—%), 
in which the parameter a has a constant 
H 











114 PLANE GEOMETRY. BOOK Il. 


value. Similarly, the variable secant PDC will be represented by the 
equation 

y¥-Y=m(x— 4%), 
in which m is a variable parameter. If one puts successively in these 
equations y = 0, x= 0, one gets the co-ordinate of the points in which 
these lines intersect the axes of co-ordinates. 


Wee | BH | Be: chee oy cee yi 

a 
B, ©=0, y=y- an, 
C, y=0, v= 1 — ¥1, 
m 
DB Os Ym i as, 


By applying the formula of § 67, one gets the equations of the lines 
AD, CB, 








x y ; 
1 = 1 
(1) pre Y1 Yr — MIX, 
a 
S y gh 
(2) now hae 
™m 


The values of x and y, which satisfy the two simultaneous equations 
(1) and (2), are the co-ordinates of the point of intersection M of the two 
lines AD and BC; these co-ordinates vary with the arbitrary parameter 
m. By subtracting the equations member from member one obtains the 
equation 








a oe ( 1 1 aa, 
eas ae ’ 
Y, — Mx, Yt Ot Y1 — MH, Yi — AX, 
or more simply 
m—a 


3 
(3) (yi — mx) (Y1 — 421) 


which, combined with equation (1), forms a system equivalent to the 
system of two equations (1) and (2). So long as the parameter m has a 
value different from a, the first factor being different from zero, the 
co-ordinates x and y of the point JZ must reduce the second factor to 
zero. Therefore the co-ordinates of each of these points of the locus _ 
satisfy the equation 





(yx + ry) = 9, 


a yix + iy = 0, 
4 Aes 
(4) a es 


This locus is a straight line passing through the origin. 

When m = a, the system of two equations (1) and (2) reduces to equa- 
tion (1); the two lines AD and BC coincide, and their point of inter- 
section is any point of the fixed secant PA. 


CHAP. III. GEOMETRICAL LOCI. 115 


Suppose the elimination had been made in another manner ; if for 
example the value of m deduced from equation (1) were substituted in 
equation (2), an equation of the second degree would be obtained, the first 
member of which would be decomposable into two linear factors of the first 
degree, and which, consequently, would represent two straight lines, the 
locus OL and the straight line PA. This equation would have the form 


(ye + my) [y — yr — 2(@ — %)] = 0. 


It is to be noticed that equation (4) does not contain the constant pa- 
rameter a; therefore the locus is independent of the particular position 
assigned to the fixed secant PA. Whence the following theorem may be 
deduced: When an angle XOY and a fixed point P, in the same plane 
are given, if any two secants PA, PC be drawn through this point P, the 
point of intersection M of the two straight lines AD and BC is always 
situated on the same straight line OL. 

Further, it is to be noticed that equation (4) depends upon the ratio 


e 1 
om that is, upon the angular coefficient of the straight line OP. Hence, 


the locus OL will remain the same, if the point P be moved along the 
line OP passing through the origin. 


104. This question can be discussed more quickly in another manner. 
Suppose that any two axes have been drawn in the plane. Represent, 
for brevity, the equations of the given straight lines OA and OB by 
a = 0, 8=0, and the fixed secant PA by y =0. The given point P will 
no longer be determined by its co-ordinates, but by the intersection of 
the two given straight lines PA and OP; the latter, passing through the 
point of intersection O of the lines OA and OB, has an equation of the 
form 8 +aa=0. The movable secant PC, drawn through the point of 
intersection P of the two lines 8 + aa =0, y =0, is represented by an 
equation of the form 


(1) B+aa.+ my =0, 


in which m represents an arbitrary parameter. The point C, in which 
this secant cuts the line OA, is given by the two simultaneous equations 
a=0,8+aa+my=0, or more simply a=0,8+my=0; the last 
equation represents a line passing through the point C, and also through 
the point of intersection B of the lines 8 = 0, y = 0; it is, therefore, the 
equation of the line BC. Similarly, the point D, where the movable 
secant intersects the line OB, is given by the two simultaneous equations 
p=0, B+aa+my=0, or more simply B=0, aa+ my =0; the line 
represented by the last equation, passing also through the point of inter- 
section A of the lines a = 0, y = 0, is none other than the line AD. The 


116 PLANE GEOMETRY. BOOK Il. 


two movable lines BC and AD, whose intersection determines the point 
M of the locus, have therefore the equations, 


(2) B+my=0, 
(3) aa+my = 0. 


The equation of the locus will be found by eliminating m between the 
two equations ; if the equations be subtracted member from member, one 
obtains the equation 


(4) B—-aa=0. 


Whence it follows that the locus is a straight line passing through the 

point O. This line is independent of y, that is, of the fixed secant PA, 

and is the same whatever be the position of the point P on the line OP. 
It has been assumed that the parameter m has a finite value; if m be 


replaced by e and after multiplying by g, one makes q = 0, the equations 
q 


(1), (2), (8) reduce to y = 0; the movable secant coincides with the fixed 
secant PA, so also the two lines BC and AD. 


105. Propiem II.— The sides of a variable triangle ABC revolve 
about three fixed points P, P', P', situated in a straight line; while the 
two vertices A and B slide on the two 
jized lines ID and IE; find the locus 
described by the third vertex C (Fig. 66). 

Draw in the plane any two axes, 
and, for brevity, represent, as in the 
preceding discussion, the equations of 
the given lines JD, JZ, bya = 0, B = 9, 
and the line PP’P"” by y = 0; each of 
the fixed points P, P’, P'’, can be de- 
fined by the intersectionrof this line 
and of a line passing through the point 
I; the point Z being the point of in- 
tersection of the lines a = 0, B = 0, the 
lines JP, IP’, IP" have equations of the form 


Bt+taa=0, B+aa=0, B+ala=0, 


in which a, a’, a!’ designate constant coefficients. In order to construct 
a particular position of the variable figure, draw through the point P an 
arbitrary line PA, then construct the lines AP! and BP"', whose inter- 
section will determine a point C of the locus. The point P being the 
intersection of the two lines y =0, B+aa=0, the line PA, drawn 
through this point, will have an equation of the form 


(1) Bt+aa+ my =), 





CHAP. III. GEOMETRICAL LOCI. 117 


in which m is an arbitrary parameter. The point A, in which the line 
PA cuts the line JD, is given by the two simultaneous equations 


a=-0, B+az+my=9, or more simply a=0, B+ my =0. 


The line AP’, passing through this point, has an equation of the form 
B+my-+mla; it is necessary to determine the coefficient m! in such a 
manner that the line passes also through the point P! determined by the 
two equations y =0, B+ aa =0; if y be put equal to zero in the equation 
of this line and 6 =—al/a, then will (m! — a@/)a= 0; asa is not zero, since 
the point P’ is not on the line a = 0, therefore must m! —a'=0, orm! =a. 
Thus the line AP’ will be represented by the equation 


(2) B+ala+ my =0. 


Similarly, the point B, in which PB cuts the line ZZ, is given by the 
two equations B = 0, 8 + da+ my = 0, or moresimply 8 = 0, aa + my =0; 
the line BP’, passing through this point, has an equation of the form 
aa+my+m'g=0; determine now the coefficient m/! in such a manner 
that this line may pass through the point P’’, the intersection of the lines 
y=0, B+ aa =0; if in this equation y be put equal to zero and 
B= —a'a, then will (a— m'a!!).a =0; therefore, choose m!! = “. 
a 
hence the line BP" is represented by the equation 


(3) an B+ ala) +my=0. 


Equations (2) and (3) are the equations of the two movable lines AP! 
and BP", whose intersection is any point C of the locus; the equation of 
the locus will be obtained by eliminating m between these two equations ; 
subtracting them member from member, one gets the equation 


(4) (a! —a)alla + (a!!— a) B=0. 


Therefore, it follows that the locus is a straight line passing through 
the point J. 


106. Corottary I. —The solution of the following problem may be 
deduced from what precedes. Inscribe in a triangle IED a second triangle 
whose edges pass respectively through the three given points P, P', P" 
lying in the same straight line. 


If a variable triangle be constructed whose sides are conditioned to pass 
through the points P’, P’, P', while the two vertices A and B slide on 
the straight lines 7D and JE, the locus of the third vertex is a straight 
line [C. The point of intersection Cj of the lines J C and DE is therefore 
one of the vertices C, of the triangle sought ; the lines C.P', C,P!' give 
the other two vertices A; and B;. It is worthy of notice that this solution 
requires the use of no other instrument than the rule. 


118 PLANE GEOMETRY. BOOK II. 


Corotiary II.—The preceding problem may be easily generalized. 
Consider a quadrilateral whose four sides pass through the four points 
P, P', Pl", Pl’ lying in a straight line, in such a manner that the three 
vertices A, B, C slide on three fixed straight lines Rk, 8S, T; find the 
locus described by the fourth vertex (Fig. 67). 








Fig. 67. 


The three sides of the triangle BCE passing through the three fixed 
points P, P’’, P’!’, revolve in such a way that the two vertices B and C 
slide on the fixed lines § and 7’; the vertex EF describes therefore a 
straight line HF. Accordingly, the three sides of the triangle AED pass- 
ing through the three fixed points P, P’, P!’, revolve in such a manner 
that the two vertices A and £ slide on the two lines R and EF; the 
vertex D describes therefore a straight line. 

From the quadrilateral one may pass to the pentagon. Moreover, 
when the n sides of a polygon pass through n fixed points lying in a 
straight line and revolve so that n — 1 of its vertices slide on n — 1 fixed 
lines, the nth vertex will describe a straight line. es 

107. Prostem III. — Being given a triangle ABA!, draw through O 
taken on the side AA! a variable secant 
OCC’; pass a circumference of a circle 
through the three points O, A, C, and a 
second through the three points O, A’, C'; 
Jind the locus of the point of intersection M 
of these two circumferences (Fig. 68). 

Take the line OA! for the x-axis and a 
perpendicular OY, drawn through O, for 
the y-axis. If @and a! be chosen as the 
abscissas of the points A and A’, the two 
fixed lines AB and A’B will have the 
equations 











CHAP. Ii. GEOMETRICAL LOCI. 119 


(1) y=c(x—a), 
(2) y=cl(e—a'), 
and the variable secant the equation 
(5) y= mx, 
in which m represents a variable parameter. The co-ordinates of the 
point Care found by solving the two simultaneous equations (1) and (3), 
which gives 
eee _ mea 
~e¢—m’ 








~C—m 
Every circle passing through the points O and A has an equation of 
the form 
v2 + y2?— ax — by =), 
in which the parameter } is arbitrary. This parameter is determined by 
the condition that the circle passes through the point C, which gives 
eo a(em +1). 
Ce ae 
the circle which passes through the three points O, A, C has therefore the 
equation 
a(cem + 1) 
Ce ee 
(4) a2 + y? — ax PURE i 
If, in this equation, a and ¢ be replaced by a! and c’, the equation of the 
circle which passes through the three points O, A’, C’ will evidently be 


: a'(c'm +1 
(5) ye? + y2? — ale — ey == 0; 


In order to find the equation of the locus of the point of intersection M 
of the two circles, it is necessary to eliminate the variable parameter m 


between the two equations (4) and (5). By equating the values of m 
deduced from (4) and (5), one gets the equation 


e(a2? + y?—ax)—ay_c(e+y?—a'xe) —aly 
(t+ y2—axr)+cay (+y—ale)+cdy 
which may be written 
(c —c!)[ (a? + y? — ax) (a? + y? — ax) + aa'y?]+ 
(1 + cc’)yfal (2? +y? — ax) — a(x? + y? — a'x)]=9, 
or (c — c!)[ (a2 + y?)2 — (a + a!)a(a? + y?) + aa!( x? + y?)] 
+ (1 + cc')(a! — a)y(x? + y*) = 0; 


by putting (2?+4 y?) without a bracket, and dividing by c—c’, one 
obtains the equation 





(1 + ec’) (a — a’) 
c—c! 





(6) (2+ y?) [2 +y—(at+a')x— y+ aa | aS 


tire 


120 PLANE GEOMETRY. BOOK It. 


This equation is decomposed into two: the one x? + y? = 0 gives the 
fixed point O in which the two variable circles intersect ; the other 


! ae 
Cea AG! a ape ae 


2 y aha ! = 
(7) x? + y*— (a+!) aoe 





is the equation of the locus of the point MZ. This locus is a circle. 

It can be seen @ priori that the three points B, A, A' belong to the 
locus. Because, if the variable secant pass through the point B, the two 
circles intersect in B; this point constitutes a part of the locus. Suppose 
now that the secant becomes parallel to the line A/B; the point C’ is 
removed to infinity, the second circle coincides with the line OA!', which 
cuts the first circle in A. In a similar manner the point A’ is found by 
supposing the secant to be made parallel to AB. It is also easy to show 
that the co-ordinates of the points B, A, A’. satisfy equation (7). Thus 
the locus required is a circle circumscribing the triangle ABA. 


108. Prosiem IV. — Being given a circle and a fixed point P, revolve 
about the fixed point Pa right angle APB; join by a straight line the two 
points A and B, in which the sides of the right 
angle produced meet the circle, and draw from 
the point Pa perpendicular PM to the line AB; 
Jind the locus of the foot M of the perpendicu- 
lar (Fig. 69). 

Take the diameter OP for the x-axis and 
the diameter perpendicular to OP for the 
y-axis ; the given circle is represented by the 
equation 


(1) x2 4+ y? = 72, 
Let (2) y=ar+b 





ft 
vc 


be the equation of the secant AB. If y be eliminated between the two 
equations (1) and (2), one gets an equation of the second degree, 


(3) (1 + a?) #2 + 2abre + 2? — 77 =0, 


whose roots are the abscissas x’ and x" of the points A and B and the 
values of the ordinates will be ax’! + b, az!’ +b. Ife represent the con- 
stant length OP, the two lines PA, PB have the angular coefficients 


! " 
y! pe eee POS Bes b axl! + ae 
go—c a“l—e xf C gl —¢ 





the angle APB being right, one has the condition 


(ax! + b)(ax!! + b) ey 
C2OC Oo 5. 





CHAP. ITI. GEOMETRICAL LOCI. 121 


which may be written 

. (1 + a%)a'a"” + (ab—c) (a +a") +0 +P°=0; 

if the values of «/+2!! and z/z!! be replaced by their values deduced from 
equation (3), one obtains the relation 

(4) (1 + a?) (c? — r?) + 2b(ac + b) = 9, 

which connects the two parameters a and b. 


The perpendicular PM, drawn from the point P to the line AB, has 
the equation 


(5) y=—2(e-0). 


The point M is determined by the equations (2) and (5), in which the 
variable parameters a and 6 satisfy equation (4); the equation of the 
locus of the point M is found by eliminating these two parameters be- 
tween the three equations (2), (4), (5). From equation (5) it follows 
y+ (&—O)% 





that a =—~-—“; whence from equation (2) one deduces b = 
On substituting these values in equation (4), one gets the equation 


(6) [yl + (ee) + yt — on + 5" =, 


which decomposes into two: the one, y?+ (« —c)? =0, gives the point 
P; the other, 
c2— 72 


(7) 2 + y?— cx + 5 a 





represents the locus sought. 

It is evident that the point P does not belong to the geometrical locus 
according to its definition ; but it is easy to understand how analysis has 
introduced it into the result. The co-ordinates « = c, y = 0 of the point 
P satisfy equation (5), whatever the parameters may be; one could 
therefore deduce from equations (2) and (4) the corresponding values of 
the two parameters a and 0; thus one finds a=+ i, b=— ae. This is 
an application of the remark made in § 102. 

Equation (7) shows that the locus is a circle having its center on the 
line OP. ‘To construct it, it suffices to determine the extremities of the 
diameter CD; if AB’, BA! be drawn making angles of 45° with the diam- 
eter OP, the chords AA’, BB’, being perpendicular to this diameter, will 
give the two points C and D. 


109. The same circle may be found by seeking the locus of the mid- 
point M’ of the chord AB. In fact, the mid-point is determined by the 
intersection of the chord AB and the perpendicular drawn from the center 
to this chord. Since these two lines have the equations 


y=ax+ b, y=— ha, 
a 


122 PLANE GEOMETRY. BOOK It. 


the equation of the locus will be obtained by eliminating the two variable 
parameters a and b between these two equations and equation (4). We 
thus have the equation 





y ey 
(+) (+ Pen t® 5 = \=0, 


which decomposes into two, the one giving the point O foreign to the 
geometrical locus, the other the circle. 


| 


110. Proptem V.—A circumference and a fixed point P are given, 
y a right angle revolves about 
its vertex placed in P; find the 
locus of the point of concur- 
rence M of the tangents drawn 
to the circumference at the 
points of intersection A and 
B with the sides of the right 
-angle (Fig. 70). 
© Take the diameter OP as 
the x-axis and the diameter 
° perpendicular to it as the y- 
axis; let r be the radius of 
the circumference and c the 
distance OP; the equation of 
Fig. 70. the given circumference is 











(1) x 4 y? = 12, 


Represent by 2; and y; the co-ordinates of any point M of the plane. 
The chord of contact of the tangents drawn from this point will have the 
equation v 


(2) eye + yy = 7. 


The co-ordinates of the points of contact will be found by solving the 
simultaneous equations (1) and (2). If 2, y be considered a solution of 
this system, the value m of the angular coefficient of the line which joins 
the corresponding point to the point P has the equation 


(3) m = —Y_ 
x—C 
The elimination of « and y from the equations (1), (2), (3) gives the 
equation which determines the angular coefficients of the two lines drawn 
from the point P to the points of intersection of line (2) with the circum- 
ference. In order to accomplish this elimination, solve equations (2) and 


CHAP. ITI. GEOMETRICAL LOCI. eke 2 


(3) for x and y, and substitute their values in (1); thus is found the 
equation of the second degree 


(4) [(v? — oxi)? + (c? — 1*) yi?) m? + 2 r?yi(c¢ — x1) m + 7° (9? — x47) = 0. 


In order that the point M, chosen arbitrarily in the plane, be a point 
of the locus, it is necessary and sufficient that the directions, which cor- 
respond to the two roots of equation (4), be rectangular. On expressing 
that the product of the roots is equal to — 1, the equation of the locus 
is found to be 

(a2 + yr?) (7? — 07) 4+ 2 rex, — 27* = 0, 


which, suppressing the indices, may be written 


2 2 4(2 72 — ¢2) 
5 ; rc ) ae as ( 
(5) (rls eee 
The locus is a circle which can be constructed by the method indicated 
in the preceding problem. 
The radii R and r of the two circumferences and the distance D 
between their centers satisfy the relation 


(6) (R? — D®)? =27°(R? + D2). 


If the sides of the right angle APB be prolonged, and tangents be 
drawn at A! and B’, the points of intersection of the consecutive tangents 
are the vertices of a variable quadrilateral, which is at one and the same 
time circumscribed about the given circle and inscribed in circle (5). 
Hence, when the radii R and r of the two circles O; and O and the dis- 
tance D between their centers satisfy relation (6), a quadrilateral can be 
constructed, inscribed in O, and circumscribed about O, by taking as an 
edge of the quadrilateral any tangent to the circle O. 








111. Propiem VI. — Find the locus of the points, such that the feet 
of the perpendiculars drawn from each of them to the sides of a triangle 
lie in a straight line. 

Let 
xcosa + ysina — p, = 0, 
(1) xcosB+ ysinB — po=9, 
xcosy+ysiny — ps3 = 9, 


be the equation of the three sides of the triangle, referred to any two 
rectangular axes, and, for the sake of brevity, represent by a1, 81, 71, the 
first members of these equations. Calling « and y the co-ordinates of the 
point M of the locus, x; and y1, Xe and yo, x3 and yg those of the feet of 
the perpendiculars drawn from the point M to the sides of the triangle, 
one has (§ 88) 


%—%=a,Cc0OSa, X—%2=—fiCospB, X—X—= V1 COSY, 


y—Yr=msine, y—Yye=fisinp, y—Yys=y18iNy. 


fa 7. ee PLANE GEOMETRY. BOOK II. 


The equation of the locus will be found by expressing the condition 
that the three points lie on a straight line. For this purpose it is neces- 
sary to equate the two ratios 2 — 4! ang 48 — 41, Tee can be put under 


x2 — Hy x — 
the form 


(ye—y)—-(n1— 9) — Ys—9)-Qi —-) 
(%2—%)—(%—%) (#3 — #)—(a1 — @) 


By substituting from the preceding equation, this equation becomes 





B6isin B —a,sin a us siny — a4 sina 
BicosB—aycosa yy ,cosy — a, COSa. 
or (2) ay; By sin (B a) + Biv1 sin (y — B) +y4a, Sin (a — y= 0. 
The letters a,, 61, 71, representing polynomials of the first es in x 
and y, it follows that equation (2) is of the second degree. The coeffi- 
cient of xy is 
sin (a + 8) sin (@— a) + sin(8 + y) sin(y — 8) + sin(y + a) sin (a — y); 
if the product of sines be transformed into the difference of cosines, this 
coefficient becomes 


(cos 2a — cos 28) + (cos 28 — cos2 y) + (cos 2y — cos 2 a) 
9 ? 


it is identically zero. The coefficients of x? and y? are 








DMs cos acos 8 sin (8—a) + cos B cosy sin (y—f8)+cos y cos asin (a — ¥), 
N = sinasinf sin (@—a)+sin B sin ysin (y—8) + sin ysinasin (a—y). 
If their sum and difference be calculated, one has 
M — .N = cos (a+ B) sin (8B — a) + cos (8 + y) sin (y — B) 
+ cos (y + a) sin (a — ¥) 
_ sin28 —sin2a+sin2y — sin28+ sin2a—sin2¥ ay, 
— 2 a med 
M+ N=cos (a — £8) sin (8 — a) + cos (8 — y) sin (y —B) 
+ cos (y — a) sin (a — y) 
_ sin2(6 — a) + sin2(y — B) + sin2(a— ¥) re 
na 2 








= — 2sin(8 — a) sin(y — 8) sin (a—¥); 
whence it follows that 
M= N= — sin (8 — a) sin (y — 8) sin (a — ¥)- 
Therefore the locus is the circumference of a circle. Equation (2) being 
satisfied when one puts 6; = y1 = 0, it follows that the point A belongs to 


the locus ; similarly with the points B and C; the locus is therefore the 
circle circumscribed about the triangle ABC. 


CHAP. IIT. GEOMETRICAL LOCI. 125 


112. From equation (2), which represents the circle circumscribed 
about a triangle whose sides are represented by equations (1), may easily 
be deduced an important property of a special system of two circles. 
Suppose that the sides (1) be tangents to a circle of radius r having its — 
center at the origin O of co-ordinates. It will be necessary in equations 
(1) to make py = po = ps3 = 7. If equation (2) of the circle be developed, 
it may be written 


(3) M (a? + y?) — Pe — Qy+ F=0. 
Let R be the radius of this circle and D be the distance of its center 
O, from the center O of the first circle ; one will have 


_ Prt 2 F praftt 2 
whence D?-— R= 
The radii of the circle O, determined by the angles a, B, y, form two 


by two the supplementary angles of the angles A, B, C of the triangle 
formed by the three tangents. One has, therefore, 


M=-—sin Asin Bsin C= -—&, 
2 #F 
F=r (sin A + sin B + sin C) = 42 cos 4 cos B cos Ole 
2 2 2 R 


S representing the area of the triangle ABC. From these results follows 


that = =— 2 Rr, and consequently 
we ; 


(4) D? = R? —2 Rr. 


Now it is proposed to determine all the triangles which are at the 
same time inscribed in the circle O; and circumscribed about the circle O, 
whose radii and the distance between the centers satisfy relation (4). It 
will be no restriction to suppose that the point O, is situated on the 
a-axis, and the angles a, 8, y to fulfill the conditions 

Pp? F P?2 
te ei Rea oe 
Se ans warn: 
But, owing to relation (4), which the given quantities R, r, and D 
satisfy, these three relations may be replaced by the two following: 


¥ 
M 


(5) Q=0, —=—2Rr. 


Let, in fact, R! be the radius of a circle circumscribed about the tri- 
angle A’ B/C’, determined by the angles a’, p’, y', which satisfy equations 


126 PLANE GEOMETRY. BOOK II. 


(5), D’ the distance of its center O! from the point O. From the preced- 
ing, it will follow 


Faroe , 
ar ae 2k'r, Di? = FR — 2 R'r. 
Moreover, by hypothesis 
— 7 =2Rr, D?= R?—2Rr; 


whence it follows that FR! is equal to R, and D’ equal to D. One of the 
three angles a’, 8’, y’, which must fulfill only the two conditions (5), can 
therefore be taken arbitrarily. Hence, when the radii R and r of the two 
circles O; and O and the distance D between their centers satisfy relation 
(4), a triangle can be constructed inscribed in Oy and circumscribed about 
O by taking any tangent to the circle O as a side of the triangle. 

The theorems analogous to the preceding and to § 110 exist for 
polygons of any number of sides. 


113. Form (2) of the equation of the circle circumscribing a triangle 
is worthy of notice. The first member has a very simple geometrical 
meaning. To be precise, suppose that the origin of co-ordinates be situ- 
ated within the triangle ABC (Fig. 71), and that 
the angles a, B, y, varying between 0 and 27, be 
arranged according to their increasing order of 
magnitude. Consider a point M having the co-or- 
dinates x and y and situated also within the tri- 
angle ; draw from this point perpendiculars to the 








ee pee ® sides, and join the feet of these perpendiculars 
<4 forming the triangle DEF. The letters a1, 81, y1 
designate the length of these perpendiculars 

Fig. 71. affected in this position by the — sign j these per- 


pendiculars are constructed in the same direction as those‘ which have 
been drawn from the origin, and which have served to determine the 
angles a, B, y. The term a, 6; sin (B—a) being equal to MD- ME. 
sin DME represents double the area of the triangle DME; the two 
remaining terms represent in a similar manner double the triangles 
EMF, FMD; thus the first member of equation (2) represents double 
the area of the triangle DEF. 

Consider next a point M’ situated without the triangle ABC. It follows 
from the figure that aj = — M'D!, 6: = — M'E', y= + M'F’; the first 
member of the equation represents double the difference between the tri- 
angle D'M E' and the sum of the two triangles Z'M!F', F'M'D'; which 
is, moreover, double the area of the triangle D/Z'F’. Whatever the posi- 
tion of the point M in the plane may be, the first member of the equation 
represents double the area of the triangle DEF affected by the + or — 


CHAP. III. GEOMETRICAL LOCI. 127 


sign. Equation (2) expresses, therefore, that the area of the triangle 
DEF is zero; that is, that the three points D, #, F lie in a straight line. 

If r be the radius of the circle circumscribing the triangle ABC, and 
d the distance of a point, whose co-ordinates are x and y, from the center 
of the circle, the first member of equation (2) can be written in the form 


AQ? + P+), 


and is equal to .A(d? — r?). This expression preserves the same sign, so 
long as the distance d is less than 7, that is, while the point M lies within 
the circle, and takes the contrary sign as soon as the point I falls without. 

It follows from the preceding that the locus of points such that the 
area of the triangle whose vertices are the feet of the three perpendiculars 
is a constant quantity k, is represented by two circles whose equations are 


ajPy sin (Bp — a) -f- PrY1 sin (y _ B) + 7141 sin (a -—— 7) =— +2 k?. 


These two circles are concentric to the circle circumscribing the triangle 
ABC: the one lies without and is always real, whatever be the given 
area; the other lies within, and is not real unless the given area is less 


than the absolute value of = : 


EXERCISES. 


1. Express the area of a triangle and of any polygon as a 
function of the co-ordinates of its vertices. 

2. Find the area of a triangle formed by lines whose equa- 
tions are given. 

3. Being given n points A, B, C,--- in a plane and n quan- 
tities m', m'', m'", --- which correspond to these n points; on 
the line AB take a point Nj, so that the distances from this 
point to the first two points are in the ratio m" to m'; then on 
the line N,C, which joins N, to the third, take a point N., so 
that its distances from the points N, and C are in the ratio 
m'" to m'+ m"; further, on the line N,D which joins the point 
N, to the fourth point D, a point N;, so that its distances from 
the points N, and D are in the ratio m! to m! +m" + ml", 
and so on, till the last given point is reached. Find the 
co-ordinates of the last point of division, which is called the 
center of proportional distances. 

When the multipliers m', m", m'" -.- are all equal to the 
same quantity, the last point of division is called the center of 
mean distances, 


128 PLANE GEOMETRY. BOOK II. 


As an application, find the quantities m’, m'', m'", which 
give, in case of a triangle, the center of gravity, the center 
of the inscribed circle, the point of intersection of the three 
altitudes, the center of the circumscribed circle. 

4. Find the locus of the points such that the sum of the 
products of the squares of the distances of each of them from 
n given points, by the quantities m', m!', m' +, 1s equal to 
a given quantity. 

5. Find the locus of the centers of circles which, viewed 
from two fixed points, subtend constant angles. 

6. Find the locus of the centers of circles which intersect 
each of two given circles in diametrically opposite points. 

7. Find the locus of points such that the sum of the dis- 
tances of each of them from two given straight lines, and in 
general from several given straight lines, is constant. 

8. Construct on two perpendicular lines OX, OY a variable 
rectangle OABC having a given perimeter 2a. Show that the 
perpendicular drawn from the vertex C to the diagonal AB 
passes through a fixed point. 

9. Being given the figure used in demonstrating the 
theorem concerning the square of the hypotenuse of a right 
- triangle, show that the two straight lines, which join the ex- 
tremities of the hypotenuse to the vertices of the squares 
constructed on the opposite sides, meet in a point on the per- 
pendicular drawn from the vertex of the right angle to the 
hypotenuse. 

10. From a fixed point P draw tangents to the circles which 
pass through two given points; find the locus of the point 
in which the chord of contacts intersects the diameter which 
passes through P. 

11. Being given a regular hexagon ABCDEF, draw the 
straight lines AC and AH; through the center draw any 
secant which cuts the two straight lines AC and AH in G 
and H; draw BG and FH; find the locus of the point of inter- 
section of these two lines. 

12. The circumferences described on the three diagonals of 
a complete quadrilateral as diameters, have two by two the 
same radical axis. 


CHAP. III. GEOMETRICAL LOCI. 129 


13. Being given five straight lines, four are chosen to form 
a complete quadrilateral, and the mid-points of its three diago- 
nals are in a straight line; the five lines thus obtained meet 
in the same point. 

14. Being given three points A, B, C and two straight lines 
X,Y; on AB as a diagonal, construct a parallelogram whose 
sides are parallel to X and Y; proceed in the same manner 
with B, Cand C, A; the second diagonals of the three paral- 
lelograms pass through the same point. , 

15. Being given four straight lines A, B, C, D, construct a 
triangle with any three and determine the common point of 
intersection of its altitudes; the four points thus determined 
lie in a straight line. 

_ 16. Two variable circles, which are tangent to each other, 
are tangent to two given circles; find the locus of the point of 
contact of the two variable circles. 

17. Four points are chosen arbitrarily on the circumference 
of acircle; the bisectors of the three pairs of angles formed 
by the lines which pass through these four points are parallel 
two by two. 

18. Find the locus of the point such that the chords of 
contact of the tangents drawn from this point to three given 
circles meet in the same point. 

19. One is given a fixed angle AOA’ and a fixed point.C on 
its bisector. An angle of constant magnitude revolves about its 
vertex placed at C’; join by a straight line the points of inter- 
section B and B' of the sides of the movable angle with the 
sides of the fixed angle and drop a perpendicular from the point 

~Cupon BB’; find the locus of the foot of the perpendicular. 

20. One is given four straight lines A, B,C, D, which taken 
three by three form four triangles. The line A belongs to 
three of these triangles; the center of the circle circumscribed 
about each of them is joined to the vertex which is not situated: 
on A; the three lines thus constructed intersect in the same 
point 7; the four points analogous to J and the centers of the 
four circles lie on the same circumference. 

21. A series of circles are given, which taken two by two 
have the same radical axis; if a variable circle cut two of these 

I 


130 : PLANE GEOMETRY. BOOK It. 


circles with constant angles, it will cut similarly each of the 
remaining circles with a constant angle. 
_ 22. The locus of the centers of circles orthogonal to two 
fixed circles is the radical axis. 

23. Show that the circle, cutting orthogonally three given 


circles 
S=9, ¢=90, y=0, 


which, taken two by two, do not have the same radical axis, is 
the locus of the points of which the polars, with respect to 
these three circles, are concurrent. 

24. Show that each of the limiting points of a pencil of 
circles and also the point at infinity on the radical axis, has 
the same polar with respect to all circles of the pencil. 


Boox Ill 
CURVES OF THE SECOND DEGREE 


——oo-@yoo— \ 


CHAPTER I 
CONSTRUCTION OF CURVES OF THE SECOND DEGREE. 


114. The general equation of the second degree between 
the variables « and y is of the form 


AT 
(1) ee ee 
A 


it involves five arbitrary parameters, the ratios of five coeffi- 
cients to the sixth. 

In order to give an account of the different forms of the 
curves which can be represented by this equation, solve it 
with respect to y. 

Two cases are to be distinguished, according as y appears 
in the equation to the second, or only to the first degree; that 
is, according as C is different from zero or equal to zero. 

Suppose that the coefficient C is not zero, and solve the 
equation with respect to y; one gets the equation 





B Dare 
(2) y= — = + GV Me + 2 Nx + P, 





by putting M= B?— AC, N= BE — CD, P= E? — CF. 
Construct the straight line DD' represented by the equation 
= Pee. 
C 
131 


132 PLANE GEOMETRY. BOOK III. 


In order now to construct the points of the locus represented 
by equation (2) for each value of a, it is necessary, starting 
from the straight line DD’, to lay off from either side along 
the ordinate a length equal to 





Y=4VMe +2 Net P. 


The line DD! (Fig. 72), which bisects the chords parallel 

| to the axis OY, is a diameter of the 
curve; the quantity Y is the length 
of the ordinate measured from the 
diameter. The construction of the 
locus is thus reduced to the study 
of the trinomial 


Me? + 2 Na+ P 


and, as the form of the locus depends 
principally on the sign of the co- 
efficient M, there will be three prin- 
cipal cases to be discussed. 








GENUS ELLIPSE. 


115. Consider the case when the coefficient M, that is, 
B? — AC, has a negative value. The ordinate is not real unless 
the trinomial has a positive value. The case investigated here 
is subdivided into three others, according to the natyre of the 
roots of this trinomial. 

1° N2?— MP>0. The two roots of the trinomial are real 
and unequal. Represent by a! the smaller, and by «" the 
larger root; the trinomial can be written 

M (x — x') (# — v"), or — M(a@ — &') (a! — 2); 
the trinomial is positive, and, consequently, the ordinate Y is 
real, for every value of a taken between the limits #' and a"; 
the trinomial is, on the contrary, negative, and the ordinate 
imaginary for every value of « less than a’ or greater than a". 

Take on the a-axis two points P’ and P" having the abscissas 
2! and a", and draw through these points the lines P!A', P!".A" 


CHAP. I. CURVES OF THE SECOND DEGREE. 133 


parallel to the y-axis; the curve will lie wholly between these 
two parallels. As the abscissa a varies from 2! to x", the ordi- 
nate Y preserves a finite value, and begins with the value zero 
and returns to zero; the locus is therefore a closed curve, which 
passes through the points A'and A", and to which has been 
given the name ellipse. | 

A value of a taken between x! and x" will be the abscissa of 
a point P situated between P' and P", and the corresponding 
value of Y will be equal to 





1 
a V(— M) PP PP". 


The variable product PP'. PP" of the two segments of the line 
P'P" is equal to the square of the rectangular ordinate of the 
circle described on P'!P" as a diameter; when the point P moves 
from P' to J, the mid-point of P'P", the rectangular ordinate of 
the circle, and consequently the quantity Y, which is propor- 
tional to it, will continually increase ; it diminishes continually, 
on the contrary, as the point moves from I to P! '. The quantity 
Y has therefore a maximum value, when the point P is at J, that 
a! + gl! N 


is, when x = — eo ae : this maximum value is equal to 


t 
(x! — : — Beginning at the point O, the middle of the 


diameter A'A", lay off along the ordinate, in opposite directions, 
a length equal to this maximum value; two points B' and B" 
of the curve will be found, and, by drawing through these 
points parallels to the diameter, a parallelogram will be formed, 
which will cireumscribe the ellipse. 

It is clear that to the two points P and Q, equally distant 
from the mid-point J, correspond equal values of Y; these 
values, laid off in opposite directions from the diameter DD, 
give the four points M, M', N, N’. The two triangles CRM, 
CSN’ being equal, the three points M, C, N' lie in a straight 
line, and the point C is the mid-point of MN '; hence all the 
points of the curve are two by two symmetrical with respect to 
the point C, the mid-point of the diameter A'A"; the point C 
is, therefore, the center of the ellipse. It follows also that the 














134 - PLANE GEOMETRY. BOOK IIt. 


lines MN, M'N' are parallel to the diameter A'A", and each 
is divided into two equal parts by the line B/B"; this line 
is a second diameter. The diameters .A’A", B'B"", each of 
which bisects the chords parallel to the other, are called con- 
jugate diameters of the ellipse. 


2° N*—MP=0. The two roots x! and a" are equal, and it 
follows that 
ee Ye = ag 
M C 
the coefficient M being negative, the quantity Y is imaginary 
for all values of # excepting w= a’, and then Y=0; the equa- 
tion has no longer a real solution, excepting the single point 
C situated on the straight line DD’. 


3° N*—MP<0. The trinomial 
: N\?. MP— N? 
x +- + (= +57) + Vv 


is negative, and consequently Y is imaginary for every value 
of x; the equation, having no real solution, does not represent 
a geometrical locus. 


GENUS HYPERBOLA. 


116. Consider next the case when the coefficient M has a 
positive value; this case subdivides into three. 


1° N?— MP>0. The trinomial 
Mx? + 2 Nu + P, 
which one writes in the form 
M (x — x')(~ — 2"), 
is positive, and consequently Y is real as a varies from 2" to 
+o, and from a to —o; moreover, Y varies at the same 
time from 0 to «. Choose, as before, on the z-axis two points 
P' and P" with the abscissas 2 and a", and draw through 


these two points the lines P!A', P''A" parallel to the y-axis; 
the curve will be situated to the right and left of these par- 


ft 
v 


CHAP. I. CURVES OF THE SECOND DEGREE. 135 


allels; it is composed of two dis- 
tinct branches extending to infinity 
(Fig. 73). This curve has been given 
the name hyperbola. If, beginning at 
the point J, the mid-point of rir, 
one lay off in opposite directions on 
the a-axis two equal lengths JP and 
IQ, the corresponding values of Y- 
are equal; the point C, the mid-point 
of A'A", is the center of the curve, 
and the two lines DD! and JC are 
two conjugate diameters. 











117. Consider the following value of y: 
_ Bet eee 3 


2 wie 
- h(a 2) 





C a; M M 
In case « has a very large numerical value, the first term of 
the quantity under the radical sign is very large as compared 
with the absolute value of the second. If the first term only 
of the quantity be considered, an approximate value of y 
will be 

Be+EH 1 N 

3 =— — —)\vV M. 
©) = Cm a(# : 7 
The preceding equation defines two distinct straight lines 
which intersect in a point of the diameter DD!', whose abscissa 





is equal to — 2; that is, equal to the half-sum of the abscissas 


of the points P’ and P"; this point is, therefore, the center C 
of the curve. Consider the branch AM of the curve; if C be 
positive, this branch is represented by the equation 


ee ce Be geri ck ie fee Ue 
eu oC (2+ 34) + rT apes 


in which allow x to vary from 2" to +0; consider at the 
same time the line CZ, which has the equation 


Ba+E 1 N 
= — — + = = Ty a 
Yi C +5(2+ i) 





136 PLANE GEOMETRY. BOOK III. 


For any value of x greater than «", the ordinate of the curve 
is less than the corresponding ordinate of the straight line; 
hence the branch AM is comprised within the angle LCD’. 
The difference y, — y of the ordinates which correspond to the 
same abscissa has the value 


n—¥= Gl (2+ g)VM—M(24 BY ead 








C M M M 
_N?—MP 1 
ee eae OF" e 
(#+ in) Vt (04-3 wy MPN 


As @ increases indefinitely, the denominator increases indefi- 
nitely, and, consequently, the difference y, — y approaches the 
limit zero. The straight line CL, which continually approaches 
the branch AM of the curve, is called the asymptote of this 
branch of the curve, which is comprised within the angle 
LCD'. Ina similar manner it can be shown that the branches 
A"M', A'N, A'N' are comprised within the angles L'CD', 
H'CD, HCD, and have as asymptotes the straight lines CL’, 
CH', CH, and each of the indefinite lines HL, H'L' is asym p- 
totic to two branches of the curve. 

It is well to notice that the angular coefficients of the 
asymptotes are given by the equation 





(4) m= —2 o vu, ¥ 
or (5) Cn? + 2 Bm + A= 0, 


which may be obtained by substituting in the terms of the . 
second degree of equation (1), 1 for x and m for y. 


118. 2° N?— MP<0. The trinomial 


N\? MP—  N? 
9 Fege AV Hak — iV" 
Mz? +2 Na+ M(2+37) + 7 


being the sum of two positive quantities, the value of Y is real 
for every value of x, and never becomes zero; Y attains its 


CHAP. I. CURVES OF THE SECOND DEGREE. 1387 


: WP_wN2 
minimum value oo when «= — * . Let I (Fig. 74) 


be the point of the x-axis whose abscissa is — + ; draw IC parallel 


to the axis OY, and take the lengths 
CB' and CB" equal to the minimum 
value of Y; the two points B'and B" 
belong to the locus. As x varies from 
i to +, or i “= to — o, 
the value of Y increases indefinitely. 
If, therefore, one draw through the 
points B' and B" parallels to the 
diameter DD', the curve is composed 
of two distinct portions, situated respectively above and below 
the parallels, and extending to infinity in opposite directions. 
The name hyperbola is also given to this curve. 








If the two values « = — “~ + abe assigned to a, and the two 


distances JP = IQ =a be laid off, starting from J, the corre- 
sponding values of Y are equal; whence it follows that the 
point Cis the center of the curve, and that the two straight 
lines DD’, IC are conjugate diameters. 


It is also easily seen that the two straight lines 


_ Be+kE 1 ae 
ee aay £G (e+ 3p) Vi 


which intersect at the center, are asymptotes of the two infinite 
branches. 


119. 3° N?— MP=0. One has then 
a NV we N 
Y=— M ——. Cae ane 
C (+5) C (+3) 


and y= tS aa 


The locus is represented by two straight lines which intersect 
on the diameter DD'. 








188 PLANE GEOMETRY. BOOK III. 


GENUS PARABOLA. 


120. Suppose finally that the coefficient Mor B* — AC be 
zero. The value of Y reduces to 


ine 7 V2 Ne + P. 
This case may be subdivided into several others. 


1° N>O. By putting — ss = x', the expression for Y may 
be written 





Y=4VING@ aoe a"). 


When 2 varies from x’ to +, the quantity Y is real and 
varies from 0 to + ©; butitis imaginary for 
all values of a less than vw’. If, therefore, 
the line P!A’' be drawn through the point P’, 
whose abscissa is «’, parallel to the y-axis, 
the curve is situated wholly to the right of 
this parallel; it passes through the point A! 
and extends, on either side of the diameter 
DD’, to infinity (Fig. 75). This curve is 
given the name parabola. 








2° N< 0. The quantity Y is real when & varies from 2! 
to — «; the curve passes through the point A’, lies wholly to 
the left of the parallel P'A', and extends to infinity ; this curve 
is also called the parabola. 


3° N=0. The value of y reduces to 


BetH 1.75 
ae aa P. 
If P is positive, this equation represents two real straight 
lines, parallel to the diameter DD’, and situated at equal dis- 
tances from this diameter. If P=0, these two parallels 
coincide with the diameter; finally, if P is negative, the equa- 
tion does not have a real solution. 


CHAP. I. CURVES OF THE SECOND DEGREE. 1389 


121. In what precedes, it has been assumed that the co- 
efficient C differed from zero. In case the coefficient C is zero 
and the coefficient A different from zero, one can solve the 
equation with respect to x and construct the locus as in the pre- 
ceding discussion; the first term of the trinomial under the 
radical has the coefficient = B’, a positive quantity or zero, 
and the locus belongs to the genus hyperbola or the genus 
parabola. . In case a variable appears in the first degree, it is 
preferable to solve the equation with respect to it; moreover, 
this method is only applicable when the two coefficients A and 
C are zero at the same time. 

It follows, by arranging equation (1) with respect to y, that 

2 (Be + EB) y+ Av’ +2Dxe+ F=0, 
_ AvP 4+2De4+F 
2 (Be + E) 

Suppose now that B be different from zero, and after arrang- 
ing with respect to the decreasing powers of a, that one divides 
till a remainder is found which does not contain a Two cases 
are distinguished, according as the remainder is different from 


or equal to zero. In the first case one will obtain a result of 
the form 





whence y= 


Cc 


—ar+b i 
y ema eae ie 


=ar+b+ 








d 8 
2 (Be + BE) 


In order to fix the ideas, assume c>0. Construct the auxil- 
iary locus defined by the equation 
: L 
y=ax +b, and put Y= or The _ 
fy — 
equation y=ax-+0 represents a M L 
straight line HZ (Fig. 76); for each 





value of x, it is necessary to increase , : 

the ordinate of this line by a quantity Lod 

QM equal to the value of Y. This oe P x 
H 








quantity becomes infinite for «= d; 
take, therefore, a point I having the 
abscissa d and draw H'L' parallel to 
OY. Ifavalue d+! be given to 2, 2’ being positive, Y will 
have a positive value, and as a! tends toward zero, Y will increase 





Fig. 76. 


140 PLANE GEOMETRY. BOOK IIt. 


indefinitely ; if, on the contrary, «x' increase indefinitely, Y 
tends towards zero; thus is obtained a curve comprised within 
the angle Z'CZ and composed of two infinite branches, asymp- 
totic to the two lines CZ, CL’. To values of a less than 
d correspond negative values of Y, and a second curve is 
obtained which is comprised within the angle HCH', and com- 
posed of two infinite branches asymptotic to the lines CH and 
CH'. Totwo equal values of x' with contrary signs, correspond 
two values of Y which are also equal and of contrary signs, and, 
consequently, two points M and M’' symmetrical with respect 
to the point C, which is the center of the curve. If the con- 
stant c were negative, one would still obtain a curve consisting 
of two distinct parts, situated in the angles HCL', H'CL. The 
curve is a hyperbola in both cases. 
If the remainder after division be zero, one has 


Av’? +2 De+ F=—2 (Bet+ EL) (ax +b), 


and the equation takes the form (y—aa— b)(Bxe+ E) =0; 
it resolves into two others, y —ax—b=0, Be + EH =0, which 
represent two lines, one of which is parallel to the y-axis. 

When A and C are zero at the same time, it is sufficient to 
put a =0 in the preceding discussion; the line DD' becomes 
parallel to the z-axis; thus is found, in one case the hyperbola 
having its asymptotes parallel to the co-ordinate axes, in the 
other two straight lines respectively parallel to the axes. 

In case the coefficients B and C are zero, the vahte of y has 
the form 7 =aa’?+4+ bxe+c; it is real whatever real value # 
may have; by causing x to vary from —o to +, one gets 
a curve which extends to infinity in two directions; this is a 
parabola. 


122. R&sum&. —In discussing the equation of the second 
degree, three species of curves have been found; closed curves, 
curves composed of two distinct parts extending to infinity 
in two directions, curves composed of a single branch extending 
to infinity in two directions. To these three species of curves 
have been given the names ellipse, hyperbola, and parabola. 


CHAP. I. CURVES OF THE SECOND DEGREE. 141 


In the beginning of this work (Book I., Chapter II.) it has 
been noticed that the curves designated by the same names in 
elementary geometry are represented by equations of the 
second degree. Conversely we shall see hereafter that all 
the curves represented by the equation of the second degree 
possess the properties which are characteristic of the definitions 
in elementary geometry, and hence that the two modes of 
definitions are equivalent. 

On reviewing the discussion, it is seen that it is the sign of 
the quantity M=B?— AC which determines the species 
of the curve represented by the equation of the second degree ; 
the curve is an ellipse, a hyperbola, or a parabola, according 
as the quantity Mis negative, positive, or zero. 

Moreover, it is important to recall that the equation does 
not always represent a curve, or what is the same, a locus; 
when the quantity M is negative, the equation represents an 
ellipse or a point, or does not admit of a real solution ; in case 
this quantity is positive, the equation represents an hyperbola, 
or two straight lines which intersect; finally, when M= 0, 
the equation represents either a parabola, or two parallel 
straight lines, or a single straight line, or it does not have a 
real solution. 


VarRIous ForMS OF THE POLYNOMIAL OF THE SECOND 
DEGREE IN TWO VARIABLES. 


123. The preceding discussion shows that the first, member 
of the equation of the curve can be put under various forms 
which it is important to characterize. Two principal cases 
are distinguished, according as C' is different from zero or 
equal to zero. 

OC 2. By solving the equation with respect to y, as 
has been done, transposing and removing the radical from 
\/ Ma? + 2 Nx + P by squaring, the equation can be put under 
the form 





(6) (Cy + Ba + E)? —(Me? + 2 Na+ P)=09. 


142 PLANE GEOMETRY. BOOK III. 


If M differ from zero (genus ellipse or hyperbola), and the 
trinomial Ma?+2Nx-+ P be resolved into a square, one has 
the form 

N\ , N?— MP 
7 Cy +B 2 —M( x) za Mera od 
(7) (Cy + Bx + E) aS) ate 
If one suppose M=0 (genus parabola), it follows from (6) 
that 


(8) (Cy + Be + E)? — (2 Nx + P)=0. 


0. 


Thus, when M ai) the first member of the equation is decom- 


posed into three squares (7), of which the last is a constant, 
these squares being affected by the signs + or —, the first 
square is affected by the + sign, the second is multiphed by 
— M, which can be positive or negative, the third can be posi- 
tive or negative. The different combinations of signs corre- 
spond to the cases which have been met in the preceding general 
discussion. They are classified in the table given below, where 
the positive square 
(Cy + Bu + E)’ 


is represented by «’, the square — MW (2 - = by + B’ or — B’, 


according as the coefficient — M is positive or negative; finally, 
the constant NSE by +? or —k’, according as it is 
positive or negative. 

When M = 0, the equation takes the form (8) of a square « 
followed by a linear function (2 Nw + P), which is represented 
by y when N is different from zero; if N be zero, this linear 
function is reduced to a constant P, which is designated by 
+ k? or — k’, according as it is positive or negative. 

Thus will arise the following table, in which it is not neces- 
sary for the moment to regard the results written in the third 
column which refer to the case C = 0 examined farther on. 

The constant c, which appears in the third column, has the 

1 = 
Wr (— AE® — FB? +2 BDE). 

This table shows that, if the inequalities M<0, and 
N?— MP < 0 exist at the same time, the equation does not 





value 


CHAP. I. CURVES OF THE SECOND DEGREE. 143 








FORM OF THE 


> ; nto J 
GENUS C20 C=0 [are EQUATION 





N2—MP>O|If C=0 the curve Real ellipse. |g?2-+ 6? k?=0 
M<0, Ellipse. |N2—MP<O} is never an Imaginary [421 g24 %2=0 





ellipse. 
N?2— MP=0 ellipse. Point. a2 af B —0 
M ' N?—MP20 cz0 Hyperbola. |q2—p24+%2=0 
>0, Hyperbola. a ee 
; m—mp=0| c=0 _| wits wuith'|a?—p2=0 


intersect. 





N20 B=0, E20 Parabola. |g2—y=—0 
Real parallel 
M=0,Parabola. |V=0, P>90/p—0 Blase straight lines, |¢?7—k2=0 
V=0, P< 9 } D2?— A F<O0|Id.imaginary.|a?-+ k?=0 
N=6: P=§.2—0 | D2 AF=0 Coincident |,2-9 


straight lines. 




















represent a locus, because the first member of the equation is then 
the sum of three squares « + 8? + k’, which cannot be zero for 
any real values of the co-ordinates: in this case the curve is said 
to represent an imaginary ellipse. Similarly, one sees at once 
that, if M< 0 and N?—MP= 0, the equation represents a point, 
because its first member « + 6, being the sum of two squares, 
can only be reduced to zero by the co-ordinates of the point 
whose co-ordinates reduce to zero at the same time these two 


squares, that is, Cy + Ba + E and +o. In case of the genus 


hyperbola, the equation represents always a locus: if 
N?— MP = 0, it represents, as has been seen, two straight lines 
which intersect; this follows at once from the preceding 
equation, because the equation, having then the form «a? — B? =0, 
decomposes into a product of two real factors of the first degree 
(« + B) (« — B) =9; itis therefore equivalent to the system of 


two equations 
at+tB=0, «—B=), 


which represents two straight lines passing through the point 
of intersection of e=0 and B=0. In analogy with this case, 
it is sometimes said, that a point-ellipse is an ensemble * of 


* The French expresses the idea better than a translation. 


144 PLANE GEOMETRY. BOOK III. 


imaginary straight lines which intersect, because the equation 
takes then the form « + £? = 0, and is algebraically equivalent 
to the ensemble of two linear equations 


a+ BV—1=0, a—BV—1=0, 


which represent nothing more than what one is accustomed for 
convenience to call the equations of conjugate imaginary straight 
lines. These two equations are satisfied by the co-ordinates of 
the point which reduce at the same time « and £ to zero, that 
is, of the point to which the ellipse is reduced. The two 
imaginary straight lines are said to intersect in this point. 


2° (=0. The curve can never belong to the genus ellipse. 
If B be different from zero, the equation can, as has been 
seen in § 121, be put under the form 





=e 
where the constant ¢, which is the remainder of the division of 
— (Av? +2 Dx+ F) 
by 2B (« + 5 has the value 
C= a _ AE? — FB’ +2BDE); 
by clearing of fractions the equation may therefore be written 
(y — ax — b)(a#—d)—c=0. “f 


The first member is a product of linear factors in w and y. 
It can also be thrown into the form of a difference of squares 
a — B°’ by writing 


y—ax—b+a—d\ _ y—ax—b—at+ pies 
2 2 


an equation of the form « — —c=0, ¢ designating a con- 
stant which may be positive, negative, or zero. 

If c¢ be different from zero, one has a real hyperbola. If 
c=0, the first member of the equation neues into the 








CHAP. I. CURVES OF THE SECOND DEGREE. 145 


‘product of two linear factors, and the curve is represented by 
two straight lines which intersect. 

If B be zero at the same time as C, # being different from 
zero, the equation becomes 


Ag? +2 Dx+2Ey+F=0. 


It represents a parabola, and can be written in the form 
a? —y=0, « and y being two linear functions, the first of 
which reduces to w If, further, Z be zero, the equation is a 
trinomial in # equal to zero, and can be written 


D\' D'— AF 
D\’_ D'— AF _¢, 
(#+%) a 


It represents two parallel straight lines real, imaginary, or 

coincident, according as D? — AF is positive, negative,.or zero. 

D?— AF 
Pr A? 

is of the form +k*% The equation takes therefore the form 

ek = 0. 


These results are given in the table on page 143; the different 
hypotheses corresponding to the case C=0 are arranged in 
the third column. 


The term « +2 is a linear function «; the constant 


Remark. —If the quantity N? — MP be constructed by re- 
placing M7, N, P by their values as functions of the coefficients 
A, B, CO, D, E, F, it follows that 


(9) N?—-MP= — C(ACF— AE’ — CD’ — FB’ + 2 BDE). 
The quantity within the parenthesis, which plays an important 


role in the theory, is called the discriminant of the curve: it is 
designated by A: 


A = AOF — AE? — CD* — FB? + 2 BDE. 


It follows from the discussion which has been given, and 
the results of which have been arranged in the preceding table, 
that the necessary and sufficient condition in order that the curve 
be a system of two real, imaginary, or coincident straight lines is 


A= 0. 


146 PLANE GEOMETRY. BOOK III. 


In fact, if neither C nor M be zero, the necessary and suffi- 
cient condition in order that the equation may represent two 
straight lines is V? — MP = 0, this is, according to (9), A= 0; 
if C be different from zero and M zero, this condition is NV = 0, 
that is, still A= 0. 

If C=0, and B be different from zero, the condition is C= 0, 
that is, by reason of the value of c, A = 0. 

If C and B are zero, this condition is H = 0, that is, still 
A 0. 


124. Seek directly the condition necessary and sufficient, in 
order that the general equation of the second degree may rep- 
resent two real or imaginary straight lines, that is, in order 
that its first member may be resolved into a product of factors 
linear in w and y: 


(10) As? + 2 Bay + Cy? +2De42Ey+F 
= (lv + my + p)(l'x + m'y + p'). 
Substitute in this identity for # and y, “and 4%, then by 
z z 


removing the denominator 2’, one will get a new identity of 
the form 


(11) ST (&, y, 2) = QR, 
where f(a, y, 2)= Ax? + 2 Bry + Cy’ + 2 Daz + 2 Eyz + Fe’, 
Q=la+myt+pz, R=l'x+m'y+ pz. 
Conversely, in case the identity (11) is given, one may return 
to the identity (10) by making z=1. Take the successive ~ 


partial derivatives of the two members of the identity (11) with 
respect to a, y, 2, then one has 


fi,= 2 (Av + By + Dz) =IR +1, 
(12) Sf, = 2 (Be + Cy + Ez) =mR+m'Q, 
fi, =2(De+ Eyt+ Fe) =prk + p'Q. 
There exists evidently at least one system of values of a, y, 2, 
2 =a, y= b, 2 =, which reduces to zero at the same time the 


linear functions R and Q, a, b, c not all three being zero. 
According to the identities (12), the same values a, 6, c reduce 


CHAP. I. CURVES OF THE SECOND DEGREE. 147 


simultaneously /’,, f',, f', to zero. Hence, when the conic is 
decomposed into two straight lines, the three linear and homo- 
geneous equations in # and y 


Ax + By + Dz =90, 
(13) Bau + Cy + Ez =0, 
De+ Ey + Fz = 0, 


admit at least of one solution « = a, y= 6, z= c, in which the 
three unknown quantities are not zero at the same time. 
Therefore the determinant of the coefficients 


ent 
ES en 2 
LS te 








is zero. This determinant is none other than the discriminant 
written above (REMARK) in a developed form. 

The condition A = 0 is therefore necessary. It is sufficient. 
In fact, suppose it fulfilled: there exists then a system of 
values a, 0, c, of x, y, 2, of which all three are not zero, satis- 
fying equations (13), that is, reducing to zero f",, f',, f’, Let, 
for example, c differ from zero: making the change of variables, 

e=az'+a', 
(14) y = bal +y/, 
Beet A 
the function f(a, y, z) will become 
FS (az' + a', bz' + y', cz’), 


that is, by developing and recalling that the function is homo- 
geneous, and of the second degree in 2, y, z, and that conse- 
quently its derivatives are homogeneous and of the first degree, 


St (&, yz) = 2" f(a, b,c) + a'z'f, + y'2'f, + Av? + 2 Ba'y'+ Cy”. 


The derivatives f’, and jf‘, are zero: f(a, b,c) is also zero by 
virtue of the identity easily verified (theorem of homogeneous 


functions), 
2f (a, b,c) = afyt+ oi+co.. 


148 PLANE GEOMETRY. BOOK ITI. 


The function f(a, y, z) is therefore identical with the 
expression Aw” + 2 Bau'y'+ Cy”, which evidently decomposes 
into the product of two linear factors (Aw’ + py’) (A'e'+ ply’), 
_and one has identically 


I (& Y, 2%) = (Aw! + py’) (A'e! + ply’). 


Returning to the variables a, y, z by aid of equations (14), 
which give 


ee a= 2 e, '— o. 
C Pi. y ¥y : 


one gets for f an expression of the form 
S (a, Y, 2) = (le + my + pz) (Ua + m'y + p'e). 


REMARK. — Designate by a, b, c, d, e, f the minors of the 
discriminant A, with respect to the elements A, B, C, D, E, 
F; thus 

a= CF—H’, b= DE—BF, c=AF-D*, 


dh OD: 6 BD An, f= AC 


The genus of the conic depends on the sign of f: when 
f is zero, the curve belongs to the genus parabola. 

From this notation, one has, on developing, the determinant 
A with respect to the elements of a row, 


A= Aa+ Bb+ Dd= Bb+ Cco+ He= Dd+ Ee + Ff. 
124. 2. As a special case, determine the necessary and 
sufficient conditions, in order that the conic be formed of two 
coincident straight lines. The first member of the equation 
is then a perfect square of a linear function of the co-ordinates, 


and one has the identity 
| (15) S (a, y, 2) = (la + my + pz)’. 


On taking the partial derivatives of this identity, it 1s seen 
immediately that the three linear equations (13) are replaced 
by a single equation, 


(16) le + my + pz = 0. 


CHAP. I. CURVES OF THE SECOND DEGREE. 149 


Their coefficients are therefore proportional, which shows 
that all the minors of A are zero. 


(17) 20 b=0; 646, 1=0, «=0, f= 6 
For example, the conditions 
A_B_D 
ys aan ed 


give, when the denominators are removed, f= 0, d= 0; etc.--- 


These conditions may also be verified directly, because the 

identity (15) gives 
A= C- a. f= p, 
D9 Sa ln, = lp, pe mp. 

It will be found on constructing the minors a, b, ¢, «++, that 
they are all zero. 

These conditions are, moreover, sufficient in order that the 
first member of the equation be a perfect square. In fact, if 
they be fulfilled, the three coefficients, A, C, F, cannot all be 
zero, because conditions (17) require that B, D, E should also 


be zero and all the coefficients would be zero. Assume then 
that A be different from zero, one will have 


Af (a, y, 2) = A’? + 2 ABay + ACy? + 2 ADxz 4+ 2 AEyz + AF?. 
Supposing that the conditions are fulfilled, one has 
AC= 3B, AE= BD, AP= UP aie 
and the preceding relation gives 
Af (a, y, 2) = (Aw + By + Cz)’. 


This subject will be considered more in detail at the end 
of Book IIL. 


150 PLANE GEOMETRY. BOOK II. 


TANGENT TO CURVES OF THE SECOND DEGREE. 


125. Let f(a, y) = 0 be the equation of a curve; if # and y 

be the co-ordinates of the point of contact M, X and Y the 
co-ordinates of any variable point of the tangent, one has seen 
(§ 89) that the tangent is represented by the equation 


(X—a)f',4+(Y—y sf, =0, 
or Xf.+ ¥f,—@f+yf) =0. | 
When the curve is of the second degree, one has 
F(a, y) = Av’? + 2 Bay + Cy? +2Dx4+2Hy+F; 
S', = 2(Av+ By + D), f',= 2 (Be+ Cy + £); 
af', + yf! = 2 (Aa? + 2 Bay + Cy? + Da + Ey). 


The point of contact M being situated on the curve, its 
co-ordinates « and y satisfy the equation. 


(1) Aa? + 2 Bay + Cy? +2 Dxe+2 Hy+ F=0. 
It follows that 
Av’? + 2 Be+ Cy =— (2 Da+2 Hy+F), 
and, consequently, 
af', + yf, =—2(De+ Ey+ F). 


The equation of the tangent, at the point whose co-ordinates 
are # and y, becomes 


(2) (Aw + By + D)X + (But Cy + E)Y¥+(Dxe+ Ey+F)=0. 


One notices that the co-ordinates x and y of the point of con- 
tact enter only to the first degree. As this equation can be 
put in the form 


(3) (AX+BY+D)e2+(BX+CY+E)y 
+ (DX+EY+F)=0, 


it is to be noticed that it does not change, in case X and a, 
Y and y are permuted. 


CHAP. I. CURVES OF THE SECOND DEGREE. 151 


It is proposed now to draw tangents to the curve from a 
given point P, not situated on the curve, and having the co- 
ordinates a, and y,; Call w and y the unknown co-ordinates of 
one of the points of contact M. These co-ordinates should 
satisfy equation (1). The tangent at the point M is repre- 
sented by equation (2). Since this tangent passes through the 
point P, the co-ordinates of this point satisfy equation (2) or 
equation (3), and one will have 


(4) (Ax, + By, + D) «+ (Ba, + Cy, + F)y+ Da, + Ey, + F=0. 


The co-ordinates a2 and y are therefore determined by the 
two simultaneous equations (1) and (4). The one being of the 
second, the other of the first degree, this system of two equa- 
tions has two solutions, and two tangents can be drawn from a 
given point P to a curve of the second degree. The solution of 
these two equations amounts to finding the points of intersection 
of the curves defined by each of them; the first is the given 
curve, the second, a straight line passing through the two points 
of contact. One notices that equation (4) of the chord of con- 
tacts has the same form as equation (2) of the tangent. It is 
sufficient to replace in the latter the co-ordinates of the point 
of contact by those of the point P. 


126. Find the condition that the straight line y=ma+k 
be tangent to a curve of the second degree. If y in equation 
(1) be replaced by mav+k, an equation will be found of the 
second degree in x, which furnishes the abscissas of the points 
of intersection of the straight line and the curve. The straight 
line will be a tangent when the two roots are equal. Thus is 
found the equation of condition 


am? —2bm +c+2dmk —2 ek + fr? =0. 
When the equation of the line has the form 
ux+vy+1=0, 
the equation of condition becomes 


(5) au? + 2 buv + cv? + 2du+2ev+f=0. 


152 PLANE GEOMETRY. BOOK III. 
The calculation may be made more symmetrical by proceed- 
ing as follows: 
Consider the hne wuX+vY+1=0, 


and suppose that it be tangent to the curve at the point whose 
co-ordinates are x and y. ‘Then the latter equation ought to be 
identical with that of the tangent at this point, and one should 
have, on representing a coefficient of proportionality by X: 


u=A(Aa+ By + D), 
(6) = v=d(Bxe+ Cy + £), 
1=d(Dx+ Ey+ F). 


By multiplying the first of these equations by a, the second 
by y, and adding to the third, one has 


ux + vy +1= (Aa? + 2 Bry + Cy +2 Du+2 Ey + F); 


since the point a, y is on the curve, the second member is zero, 


and one has 
ux +vy+1=0, 


or rA (ux + vy + 1) = 0. 


This equation, combined with equations (6), gives a system 
of four equations of the first degree in Aa, Ay, and A. The 
elimination of these three quantities gives the condition sought 
in the form of a determinant: 


A B D U 
B C E v 
D E F 1 
U v 1 Gi=0, 








whose development leads to equation (5). 


CHAP. Il. CENTER, DIAMETERS, AND AXES. 153 


CHAPTER II 


CENTER, DIAMETERS, AND AXES OF CURVES OF THE 
SECOND DEGREE. 


127. The center of a curve has been defined as a fixed point 
C, with respect to which all the points of the curve are sym- 
metrical two by two. In the discussion of the general equa- 
tion of the second degree, it was found that the ellipse and 
hyperbola have a center. It is proposed now to determine 
directly the center of a curve of the second degree without 
solving the equation. The method which will be used depends 
on the theorem: when the origin of co-ordinates is the center 
of a curve of the second degree, the equation of the curve does 
not contain the terms of the first degree. 


Let 
(1) Ax? + 2 Bay + Cy? +2Dxe4+2Ey+F=0 
be the equation of a curve of the second degree having the 


origin at the center (Fig. 77); the equation of a straight line 
MM' drawn through the origin 


has the form y=mza. The elim- _ ‘Yy 
ination of y between this equa- ne 
tion and that of the curve gives 
the equation . > 
(2) (A+2 Bm + Cm’) # iy 
+2(D+ Em)x+ F=0, ; 
0 


a 








which determines the abscissas 
of the two points of intersec- 
tion. The origin being the mid-point of the line MM’, the’ 
preceding equation ought to have equal roots with contrary 
signs, and this will be the case if the coefficient of the first 
power of 2 be zero; one has, therefore, D+ Em=0, and, 


Fig. 77. 


154 PLANE GEOMETRY. BOOK III. 


since this condition should hold for an infinity of values of 
m, one should have separately D=0, H=0. Conversely, 
when these conditions are fulfilled, equation (2) has two equal 
roots with contrary signs, whatever be the value of m, and, 
consequently, the origin is the center of the curve. 


128. In order to know if a locus of the second degree have 
a center, keeping the axes parallel to themselves, transfer the 
origin to an arbitrary point whose co-ordinates are a and J, 
then examine whether these quantities can be so determined 
that the new equation does not contain the terms of the first 
degree. 

The formulas for transferring the axes parallel to themselves 
arex=a+e', y=b+y'. On substituting in equation (1), 
the new equation will be » 


(3) Aa?+ 2 Ba'y'+ Cy?+ 2(Aa+ Bb + D)2'+ 2(Ba+ Cb + E)y' 
+Ad?+ 2 Bab+ Ce? + 2Da+2 H+ F=0, 


whose composition should be carefully noted. Represent, for 
brevity, the first member of equation (1) by f(a, y), which is 
an integral function of the second degree in # and y; in equa- 
tion (3), the terms of the second degree are the same as in 
equation (1); the terms of the first degree have as coefficients 
the partial derivatives of the function f(a, y), taken with respect 
to the variables # and y, and in which the variables have been 
replaced by a and 6; finally, the constant term is the value 
which the polynomial f(#, y) takes for «=a and ¥'=), and 
equation (2) can therefore be written 


(4) Ax? + 2 Ba'y'+ Cy” + fl.(a, b)x' + f%, (a, b)y' + f(a, b)= 0. 
On equating to zero the coefficients of x' and y', one obtains 
the two equations of the first degree, 
Aa+ Bb+ D=0, 
(5) 
It follows, therefore, that the center of a curve of the second 
degree is determined by solving the two equations which are 


found by equating to zero the partial derivatives of the first 
member of the given equation, taken with respect to x and y. 


CHAP. Il. CENTER, DIAMETERS, AND AXES. 155 


129. If a and bd be regarded as variable co-ordinates, each of 
the equations (5) defines a straight line, and there is occasion 
to distinguish several cases, according as the denominator, 
common to the values of the unknown quantities, or the 
determinant AC — B’, which has been represented by — M or 
f, is different from zero or equal to zero. 


1° When the determinant f is different from zero, the sys- 
tem of equations is satisfied by one system of values of a and b 
and by one only; the two straight lines intersect; the curve 
has a center and a definite center, whose co-ordinates are, accord- 
ing to § 124, 


cS . 
6) : 
f 


oe 


2° In case the determinant f is equal to zero (genus parab- 
ola), the lines are parallel, or coincide. In the first case, the 
curve does not have a center; in the second case, every point 
of the straight line defined by one of equations (5) is a center. 
It is easy to see that, in the latter case, the locus, if it exist, 1s 
necessarily composed of two parallel sea lines. Let, in 
fact, CC' be the straight line which is 


the locus of the centers (Fig. 78), and ae we 
M a point belonging to the locus; join We 
i. 


the point M to the various points of  ¢ W r\ 5 








the straight line CC’, and prolong each 
of these straight lines till JN is equal - oT 
to IM, etc. The points N, N’, N", ---, 
thus obtained, will belong to the locus. 
Now all of these points are situated on a line parallel to CC". 
Proceeding in the same manner with the point N, a second 
parallel MM’ will be determined. Moreover, equation (1) can- 
not represent other points than those of these straight lines ; 
otherwise a straight line could intersect the locus in more than 
two points. If the point M were situated on the line CC’, the 
two parallels would coincide with the locus of the centers. 








Fig. 78. 


156 PLANE GEOMETRY. BOOK III. 


130. If the curve have a center, and the origin be trans- 
formed to this point while the axes remain parallel to them- 
selves, the equation simplifies and becomes 


(7) | Ax” + 2 Bo'y'+ Cy? + H=0, 


since the terms of the first degree disappear. The constant 
term H of the new equation has the value 


H= Ad’+ 2 Bab+ Cl?+2Da+2 Eb + F, 


a and b representing the co-ordinates of the center. But the 
quantities a and 6 satisfy equations (5): if the members of 
each of them be multiplied respectively by a and b, and added 
together, they become 


Ad? + 2 Bab + Cb? + Da + Eb = 0, 
whence Aad? + 2 Bab + Cb? = — (Da + Eb), 
and, consequently, H= Da+ Eb+4+ F; 
and by replacing a and b by their values (6) 





(8) ii 7 
When the discriminant A is zero, the equation reduces to 
(9) Ax” + 2 Bu'y'+ Cy” =0, 
whence . 
—B+VB?— AC 
(10) Jie G a! 


If the quantity B’— AC be negative, the equation has the 
real solution 2’ =0, y'=0. If it be positive, the equation 
represents two straight lines passing through the origin. In 
this case equation (7), in which any arbitrary value may be 
assigned to H, defines a hyperbola; it has been found (§ 117) 
that the asymptotes of a hyperbola pass through its center, 
and that their angular coefficients are given by the formula 
m= BANE AC AC, 


C 


these asymptotes are none other than the straight lines repre- 
sented by equation (10) or by equation (9). Thus, when an 





CHAP. Il. CENTER, DIAMETERS, AND AXES. 157 


equation of the second degree represents a hyperbola referred 
to its center, the equation of the asymptotes is found by sup- 
pressing the constant term in the given equation. 

From this it follows that if the general equation of the 
second degree, 


f (a, y) = Aa? + 2 Bay + Oy +2 Du+2 Hy + F=0, 


represent a hyperbola, the equation 


(11) f(@ y-F=0 
represents the ensemble of two asymptotes. In fact, if the ori- 
gin be transferred to the center of the curve, f(a, y) becomes 
Av” + 2 Baly' + Cy” + 2; 
therefore, equation (11) becomes 
Az”? + 2 Ba'y'+ Cy” =), 


which is the equation of the asymptotes. 


DIAMETERS. 


131. If acurve of the second degree be cut by a system of 
parallel straight lines, the locus of the mid-points I I of the 
chords MM", determined by the two points of intersection, 1s 
a diameter of the curve. Let m be the angular coefficients of 
the chords, and 


(1) f(x, y) = Aa’ + 2 Bay + Cy’ +2 Dxe+2 Ey+ F=0 


be the equation of the curve. If the axes be kept parallel to 
themselves, and the origin be transferred to the arbitrary point 
T of the plane, whose co-ordinates are a and 8, the equation of 
the curve becomes (§ 128) 


(1) Av? £2.Be'y! + Cy? +1 (ab) 2! +f5(a, Dy! + £(@0) = 


Draw through this point J a line MM" parallel to the given 
direction; the equation of this parallel is y'=ma'. The 


158 PLANE GEOMETRY. BOOK Itt. 


elimination of y' between this equation and that of the curve 
leads to the equation of the second degree, 


(1) [A+2Bm+Cm*] x”? + (f(a, b) +f, (a, b) m]a! 
+f (4, b) =9, 

which gives the abscissas of the points of intersection. So 
long as the value assigned to m does not 
reduce A+2Bm+Cm?, the coefficient 
of x’, to zero, each of the secants inter- 
sects the curve in two points; if it be 
assumed that the origin J be placed at 
the mid-point of the chord MM’ (Fig. 
79), equation (11) having its roots equal 
with contrary signs, one has the relation 


(12) t', (a, 6) + f(a, b)m=0 


this equation being satisfied by the co-ordinates of the mid- 
point of any of the chords considered, is the equation of the 
locus. If aand b be replaced by x and y, it becomes 





Fig. 79. 


(13) S'2(@Y) + mf, ( y) = 9, 
or 
(14) (Aw + By + D) + m (Be + Cy + E) = 0. 


Since this equation is of the first degree, it follows that the 
diameter, which corresponds to any system of parallel chords, 
is a straight line DD'. Call m' the angular coefficient of the 
diameter; we shall have the relation ? 


% ' A+ Bm 
-) ie oes 
or 
(16) Cmm'+ B(m+m') + A= 0. 


132. Remark I.—The values of # and y, which satisfy the 
simultaneous equations 
Az+ By+ D=0, Beu+ Cy+ H=0, 


satisfy equation (14), whatever be the value of m; therefore 
if the locus have a definite center, all of the diameters pass 
through the center, and, if it have an infinity of centers, all of 
the diameters coincide with the locus of the centers. 


CHAP. Il. CENTER, DIAMETERS, AND AXES. 159 


The two equations, which determine the center, represent 
two diameters; the first corresponds to the chords parallel to 
the a-axis, the second to chords parallel to the y-axis. They 
are formed by putting m = 0 or m= o. 


133. Remarx II.—In case the curve be an ellipse, the 
trinomial A+ 2 Bm + Cm’, which has imaginary roots, is always 
different from zero; to every direction of the chords cor- 
responds a diameter defined by equation (14). 

This equation (14), if m be regarded as an arbitrary param- 
eter, represents all of the straight lines which pass through 
the center; whence it follows that every straight line passing 
through the center is a diameter. 


Remark III.—In case of the hyperbola, the trinomial 
A+2Bm + Om? becomes zero for two real values of m which 
are the angular coefficients of the asymptotes. If one of these 
values be given to m, equation (11) is depressed to the first 
degree; each of the secants intersects the curve in but one 
point. If, further, the co-ordinates a and 6b of the point J, 
through which the secant is drawn, satisfy relation (12), equa- 
tion (11), having its first two coefficients zero, has no longer a 
solution; the straight line represented by equation (14) is then 
the locus of the points J such that the parallels drawn through 
each of its points with the given direction do not intersect the 
curve; but, by reason of relation (16), the value of m! being 
equal to m, all of these parallels coincide with line (14) itself. 
Since this line passes through the center, it is one of the 
asymptotes. 

If m be regarded as an arbitrary parameter, equation (14) 
represents all of the straight lines which pass through the 
center; whence it follows that all of these straight lines, except- 
ing the two asymptotes, are diameters. 


134. Remark IV.—In case of the parabola, we have 
AC — B’= 0, or <= a3 whence it follows that the value of 


m', given by equation (15), is independent of m and equal to 


160 PLANE GEOMETRY. BOOK III. 


oe ai thus all of the diameters of the parabola are parallel to 
each other. 

The trinomial A + 2 Bm-+ Cm? has its two roots equal to 
— a the angular coefficient of the diameters. If parallel 


secants be drawn in this direction, each of them will intersect the 
curve in but one point. On the other hand, if the value -2 
be assigned to m, the coefficients of # and y in equation (14) 
become zero and the equation ceases to represent a straight 
line. | 


Equation (14), in which m is regarded as an arbitrary 
parameter, represents all of the straight lines parallel to the 


direction rei whence it follows that every straight line 


parallel to this direction is a diameter of the parabola. 
If at the same time AC— B?=0 and BE —CD=0, 


or “ =5> a , the locus of the second degree is represented 
by two parallel straight lines; if — m! represent the common 
value of the preceding ratios, 


Az + By + D=—m' (Be + Cy + B), 
and equation (14) reduces to 
(m — m') (Bu + Cy + E)=09. 


Thus, in this case, all of the diameters coincide. 


CONJUGATE DIAMETERS. 


135. Assume that AC — B’ differs from zero. The two 
coefficients m, and m' are connected by the relation 


(16) Cmm'+ Bim +m') + A=0. 


Imagine secants to be so drawn that the chord MM" be 
parallel to the diameter DD' (Fig. 79); let m' be the angular 
coefficient of the diameter HE' which bisects these chords, 


CHAP. Il. CENTER, DIAMETERS, AND AXES. 161 


then the relation between the direction m' of the chords and 
the direction m" of the corresponding diameter HE’, will be 


Cm'm" + B(m'+m") + A=0; 


this and the preceding equation being of the first degree with 
respect to m!' and m, it follows that m'' = m. The two diameters 
DD' and EE", whose angular coefficients are m!' and m, have the 
property, that each of them bisects the chords parallel to the 
other; they are for this reason called conjugate diameters. 

The ellipse and the hyperbola have an infinity of systems of 
conjugate diameters. One can take for the first diameter any 
straight line drawn through the center, provided that, if the 
curve be a hyperbola, it does not coincide with one of the 
asymptotes. 


136. It has been seen (§ 130) that the equation of the 
curve, referred to axes parallel to the primitive axes and drawn 
through the center, is 


(17) Aa? + 2 Bay + Cy? + H= 0. 


If any two diameters be taken as axes of co-ordinates, and as 
this transformation may be accomplished by aid of formulas (4) 
of § 51, the homogeneous polynomial of the second degree 
Ax? + 2 Bey + C7’ transforming into a homogeneous polynomial 
of the second degree A'x” + 2 Bla'y'+ Cly”, the equation 
becomes 
Alu” + = B'a'y' + C'y” + H= 0. 

In case the two diameters are conjugate, since to each value of 
x' two equal values of y' with opposite signs correspond, the 
coefficient B' is zero, and the equation reduces to the simple 
form 


(18) A'x” + Oy" + H=0. 


In case of the parabola, if a point on the curve be taken as 
the origin, which causes the constant term to vanish, the 
diameter which passes through this point as the a’-axis, a line 
through this point parallel to the chords which the diameter 


bisects as the y'-axis, since to each value of x! correspond two 
L / 


162 PLANE GEOMETRY. BOOK III. 


equal values of y' with opposite signs, one ought to have 
B'x' + E' = 0, and, consequently, separately B' = 0, E'=90; on 
the other hand, since the curve is a parabola, the eanaigion 
A'C' — B” = 0 ought to be satisfied, which gives A'=0; thus 
the equation reduces to the simple form 


(19) Cly® +2 D'z' = 0. 


From which it is seen that the y'axis coincides with the 
tangent at the origin. 


AXES. 


137. In curves of the second degree, the diameters perpen- 
dicular to the chords which they bisect are the aves of sym- 
metry. 

The parabola having all its diameters parallel, if one imagine 

ee series of chords MM’ (Fig. 80) perpen- 











a a dicular to the common direction of the 

diameters, the diameter AA’, which bisects 

a the chords, will be an axis of the curve and 

3 a it will be the only one. The angular co- 

N efficient of the diameter is — - therefore, 
—~ | Cc’ 

~~ f the co-ordinates are rectangular, the axis 


Fig. 80. of the curve is the diameter of the chords 


having the angular coefficient ‘ ; its equation (§ 131) is 


(20) B (Aa + By + D) + C (Be + Cy + £) = 0. 


In case of oblique co-ordinates, the angular coefficient of the 
chords perpendicular to the axis being 


C— Bcos@ 
B—Ccos@ 





this straight line is determined by the equation 
(da + By + D) (B—Ccos 6) + (Ba + Cy + E) (C—Beos 6) =0. 


The equation of the parabola referred to its axis AA!’ and to 
the tangent at the vertex A, is of the form (19). 


CHAP. II. CENTER, DIAMETERS, AND AXES. 163 


When the curve is an ellipse or a hyperbola, referred to the 
axis AA! (Fig. 81), and a second BB' cor- 
responding to it, forming with the first a es 
system of conjugate diameters, the ques- ( 
tion is then reduced to finding a pair of — - 
perpendicular conjugate diameters. If y 
the co-ordinates be rectangular, the angu- ae A 
lar coefficients of the axes are found by 
combining the relation mm!=—1 with be 


C—A 
B 














equation (16) which gives m-+m!'= ; thus, m and m’ 


are the roots of an equation of the second degree, 
(21) Bw +(A— C)u—B=0. 


Should the origin of co-ordinates coincide with the center, 
the equation 


(22) By? +(A — C) ay — Ba’ = 0, 


which is deduced from equation (21) by substituting © Y for u, 
represents the ensemble of the two axes. 

In case of oblique co-ordinates, the angular coefficients of 
the axes are the roots of the equation 


(23) (B—Ccos6)w+(A— C)u—(B—Acos6)=0. 


The equation of the curve, referred to these two axes, is of 
the form (18). 

Let u be a root of one of the equations (21) or (23); the 
equation of the corresponding axes will be 


f+, =0. 


Therefore, the origin of co-ordinates being chosen in any 
manner, one will have an equation of the second degree repre- 


Ss 


senting the ensemble of the axes on replacing wu by —*,, in 
y 
(21) or (23), which gives, when the axes are rectangular 


(eq. 21), 
(24) Bf —fy)—(A— O)P fy = 9. 


164 PLANE GEOMETRY. BOOK III. 


137. 2. To determine the position of a point with respect to 
@ conte. 

Let f(x, y)= Ax’? +2 Boy + OP+2Dxe+2 Hy +F 
be the first member of the equation of a conic. If the point 
M (x, 7) be displaced in the plane in a continuous manner by 
making it follow any arbitrary path, the function f(a, y) varies 
in a continuous manner, and can only change sign when it 
becomes zero; that is, when the point M crosses the curve. 
The sign of f(a, y) is therefore the same for all points of the 
plane situated on the same side of the curve; moreover, this 
sign changes when the point M crosses a simple branch of the 
curve. In fact, let y= mau+h be the equation of a secant 
M'M" cutting the curve in two real distinct points 1’, M", 
with the abscissas v7’ and «''; displacing the point M on this 
secant, one will have 


S(&, Yy=Sf(e, mx +h); 

the function /(x,ma-+h) is a trinomial of the second degree 
in a having the roots #' and w'. The trinomial has a certain 
sign when 2 is situated without the interval xx", and the oppo- 
site sign when a hes within this interval. Therefore, when 
the point M is displaced on this indefinite straight line M'M", 
the function f(a, mx + h), or its equal f(a, y), has a certain sign 
so long as the point M is exterior to the segment M'M", and 
the opposite sign when the point is on this segment. The 
sign of f(a, 7) changes, therefore, when the point ,M crosses 
the curve on a secant. 

Accordingly, it suffices to know the sign of f(a, y) for one 
point of the plane not situated on the curve in order to know 
its sign for any portion of the plane. Take, for example, the 
function 

S(®, Y= ve? + ay ty —2e+y, 

which, equated to zero, represents a real ellipse. If one take 
a point M(a, y), situated at a sufficient distance, it will be 
exterior to the curve; take, for example, on the axis Oy (a = 0, 
y sufficiently large); then f(a, y), which is reduced to a tri- 
nomial in y, is evidently positive. Therefore, in this example, 
J (x, y) is positive without the ellipse, and consequently nega- 


CHAP. Il. CENTER, DIAMETERS, AND AXES. 165 


tive within. One can perceive immediately the position 
of a point with respect to the curve from the sign of 
S (®, Y)- 

We come now to the general case, and seek to give simple 
rules for the different cases. 


1° If the equation f(«, 7)=0 represent an imaginary ellipse 
or a point-ellipse, or two imaginary parallel straight lines, or 
two coincident straight lines, the function f(@#, y) has the same 
sign for every point of the plane; because, in case of the three 
first hypotheses, it reduces to zero for but one point at most, 
and in the last (coincident straight lines), the function f(a, y) 
is a perfect square. 


2° If f(a, y)=0 represent a real ellipse or a hyperbola, it is 
convenient to call the region of the plane which contains the 
center the interior of the curve; the remainder of the plane, 
the exterior. The signs of f(a, y) are different for the exterior 
and the interior. Let aand b be the co-ordinates of the center, 
the function f(a, 7) takes for the center the value (§ 130) 


A A 
1G at Am 
the sign of this quantity furnishes therefore the sign of 
f(x, y) for the interior of the conic; the sign will be the > 
opposite for the exterior. 


3° If f(#, y)=9 represent a parabola or two real parallel 
straight lines, the interior of the curve is the region which 
contains the focus of the parabola or the region comprised 
between the two straight lines. The sign of f(a, y) may be 
obtained immediately for the exterior of the curve by taking 
the sign of f(a, y) for a point at infinity in a direction not 
parallel to the direction of the axis or to that of the two lines. 
This particular direction is obtained by equating to zero all of 
the terms of the second degree Aa? + 2 Bry + Cy’, which is a 
perfect square in the case considered. Since one of the two 
co-ordinate axes at least is not parallel to this particular 
direction, it will suffice to take the sign of f(#, y) for infinity on 


166 ; PLANE GEOMETRY. BOOK III. 


this co-ordinate axis; that is, the sign of the coefficient A or C 
which is not zero. 


4° If the curve f(a, y)= 0 be composed of two straight lines 
which intersect, the sign of f(2, y) will be the same in the 
vertical angles; the signs of f(a, y) are opposite in the adjacent 
angles. 


CHAP. Ill. REDUCTION OF THE EQUATION. 167 


CHAPTER III 


REDUCTION OF THE EQUATION OF THE SECOND 
DEGREE. 


138. In order to study with the most facility the properties 
of a curve of the second degree, it is important to simplify as 
much as possible its equation by referring it to co-ordinate axes 
suitably chosen. It has been seen, in the preceding chapter, 
that the equation of the second degree can always be reduced 
to one of the two forms 


(a) A? + Oy+H=0, (f) Cy? + 2 Dx = 0. 


In case the curve is an ellipse or a hyperbola, its equation 
is reduced to the form («), on taking any two conjugate diame- 
ters for a system of co-ordinate axes; in general, the co-ordi- 
nates will be oblique; they will be rectangular, if the curve be 
referred to its axes. In case the curve is a parabola, its equa- 
tion is reduced to the form (8), on taking any diameter as the 
axis of a, and a tangent at the extremity of this diameter as 
the axis of y; in case the co-ordinates are rectangular, one 
takes the axis of the curve as the a-axis. 

It is by means of these equations (a) and (Q), in rectangular 
co-ordinates, that one demonstrates in most part the properties 
of the curve of the second degree. One applies now the method 
used to accomplish the reduction of the equation. Let 


(1) Ag? + 2 Bay + Cy +2 Dxe+2Ey+Fr=0 


be the given equation of the second degree referred to rectan- 
gular axes; if they were not, one would first render them such 
by a transformation. On retaining the z-axis and taking for 
the y-axis the perpendicular erected to the z-axis at the origin, 
the formulas of transformation are 





! 
rae C08 O Ma 


e = & ; =. 
sin 6 sin 6 


168 PLANE GEOMETRY. BOOK IU. 


ELLIPSE AND HYPERBOLA. 


139. Consider now the case where the quantity AC — B? is 
different from zero; the curve has a definite center, whose 
co-ordinates a and } are given by the formulas (§ 129) 

At ee) 


a=-) 
f : 





Keeping the axes parallel to them- 
C * selves, transfer the origin to the 
BI center C (Fig. 82). One knows that 

A the terms of the second degree 

<a es = ~~ do not change, that those of the 
ara first degree disappear, and that 

the constant term H of the new equation is given by the 











formula = The equation of the curve, by this change of 
co-ordinates, simplifies and becomes 

(2) Ax? 4 Z Bay, + Cy? ob A= 0. 

Rotate now the co-ordinate axes, supposed rectangular, about 
the center C through the angle «, in order that they may coin- 


cide with the axes of the curves. The formulas of trans- 
formation are 


%=2'cosa—y'sina, y,=2'sina+y! COS ¢t. 
Substituting in equation (2), one gets the new equation 
(3) (Acos’« + Csin?« + 2 B sin a cos «) a” 
+ (A sin?a + Ccos?a — 2 Bsin a cos a) y" 
+ 2[(C—A) sin « cos a + B(cos? a — sin? «)Ja'y'+ H = 0. 


The angle « may be so determined that the coefficient of the 
term a'y' will be null; for this purpose, one will put 


(4) (C' —A) sina cosa +B (cos” « — sin? a) = 0, 
or 
(5) Btan?a+(A—C)tana—B=0. 


CHAP. Ill REDUCTION OF THE EQUATION. 169 


This equation of the second degree is the same as equation 
(21), § 137, by which the directions of the axes of the curve 
are determined. Equation (4) can be solved more simply by 
putting it under the form 


(C —A) sin2a+ 2 Becos2a=0, 
whence 


(6) fg a ee 


A-—C 

If the case of the circle be excluded, where one has at the 
same time B= 0 and A =C, equation (6) gives for 2«@ a posi- 
tive value w less than z, and the various values of 2 which 
satisfy this equation are represented by the formula 


2Za=o+kn, 


where k designates any integral number positive or negative; 


whence one deduces = ; se ke 


The different values of « furnish no more than four different 
directions for the axis C:X'; these four directions are two by 
two opposite, and determine two perpendicular straight lines. 


One gives to « the value S which is always positive and less 


than =. 
an 5 


140. The term in a'y' disappears from equation (3); it 
remains to calculate the value of the coefficients of the terms | 
inv? and y”. If one put 


A'=A cos?a+Csin?a + 2B sina cos a, 
C'=Asin?a+Ccos*’« —2 Bsin« cos a, 
one will have 
A'+C'=A40C, | 
(7) A'—C'=(A —C) (cos? « — sin’ «) +4 Bsin «cosa, 
=(A —C) cos2a+2Bsin2«. 


170 PLANE GEOMETRY. BOOK IIt. 


Equation (6) gives 
he » cos2a= ee 
+ V4 B+ (A—CY 


sin2a= 














---VEBFy (4—OF 
it follows A'— C'=+VEBF +(4—CYy. 





The two coefficients A’ and C! can be calculated by means of 
the formulas 





! a 
(8) A'+C'=A4+C, 
A'—C'=R, 
on putting R=V4 B+ (A—C)* 


The value of 2a was taken positive and less than 7. Sin2«a 
having a positive value, it will be necessary to give the radical 
the sign which B has. In this manner the equation of the 
curve is reduced to the simple form 


(9) Alc? + Oly? + H=0. 
It represents an ellipse or a hyperbola, according as the two 
coefficients A’ and C’ have the same or opposite signs. 


The preceding formulas (8), squared and subtracted, furnish 


the relation 
A'C' = AC — B’. 


The coefficients A' and C’ of the equation of the curve referred 
to its axes are the roots of the equation 


(10) S?— (A+ C)S+ (AC— B’) =0. 
The dimensions of the curve defined by equation (9) depend 


t 
vc 


os a In case of the ellipse, 
these two quotients, which have the same sign, are positive; if 
they be represented by a? and b’, a and 6 will be the segments 
of CX' and CY' comprised between the center and the curve. 
The lengths 2a and 20 are called the axes of the ellipse. The 
quantities a? and U? are the roots of the equation of the second 
degree 

(11) (AC — B’)wW+ (A+ C) Hu + H’=0, 


i 
on the two parameters — 


which is obtained by substituting — FT soy § in equation (10). 
U 


CHAP. III. REDUCTION OF THE EQUATION. 171 


In case of the hyperbola, the two quotients -3, = a have 
opposite signs. If they be represented by a and — b%, or — a 
and 67, according to the two cases which can occur, these two 
quantities are still the roots of equation (11). The quantities 
2a and 20 are called the axes of the hyperbola. 


PARABOLA. 


141. When AC — B’? = 0, the terms of the second degree in 


the given equation form a perfect square. One has, in fact, 
2 


on replacing A by its value a 

2 2 

Ax? + 2 Bay + Cy = o(v +B oy + Ge) = fy +") 
and the equation can be written 


O(y+ Ge) +2De+2 Ey + P= 0. 


Rotate the axes of co-ordinates about the origin through an 
angle « (Fig. 83) by means of the formulas of transformation, 


v= 2, COSa—Y, SING, y= Sina + ¥ COSa4; 


the proposed equation becomes 


2 
(12) o| (cos «— 2 sin «) n+ (sin «+ 2eos «) n| 


+2(Deosa+ Esina)2,+2(Heosa— Dsina)y, + F=0. 
One can determine the 
value of « so that the co- 
efficient of 2, or y 18 zero x 
in that part of the poly- Se a 
nomial which is squared. y, 
Put, for example, 


iy 


. B 
sin « + 700s a = 0; 





whence ag 


? 





(13) tan a = — 


alo 


IY &- PLANE GEOMETRY. BOOK IIt. 








then equation (12) will simplify and become 


(14) Cyr +2 D'x, +2 E'y,+F=0, 
where 
C=C cosa —Gsina) = C(cos « + tan @ sin a)? = ¢ ; 
C COs? & 





and, consequently, 
2 2 
=F t Ate, 





The coefficients D' and H' are obtained by replacing sin @ and 
cosa by their values, which give 


pi-_CD-BE »,__CE+BD 


SE OO). +VO(A+ 0) 














One of the values of « given by equation (13) is positive 
and less than +. If one take this value, sin @ will be positive, 
and it will be necessary to give the radical a sign opposite to 
that of B. If the coefficient D' were zero, equation (14) would 
no longer contain 2,, and would represent two straight lines 
parallel to the axis OD,. Incase this coefficient is different from 
zero, one transfers the axes parallel to themselves by putting 


%=a+2', yW=b+y'; . 
equation (14) becomes 
Cy? +2 D'iz' +2(C'b + FE") y'+ (CV? +2 Dia+2 H'b + F)=0. 


The co-ordinates of the new origin A are so determined that 
the coefficient of y' and the constant term are zero, 


Cb + E'=0, CU +2Da4+2Eb4+F=0, 


which give finite values for a and b, and the equation will 
reduce to the simple form 


(15) O'y” +2 D'x! = 0. 


CHAP. Ill. REDUCTION OF THE EQUATION. 173 


The dimensions of the curve depend on the numerical value 
! 
of the quotient o or of that of 


BE — CD : 
(A+C)VC(A+C) 
This quantity is called the parameter of the parabola. 





142. The coefficients of the reduced equations can be easily 
calculated by employing certain functions of the coefficients 
and of the angle between the axes, which do not change in 
value when any change whatever of the axes is made. To © 
form these functions, take formulas (4) of § 51: 
2 — 2 sin (@— a) + y'sin 6 — B), 


sin 6 





(16) 


_a@'sina+y'sinB 
sin 0 








which serve for the transformation of co-ordinates, when the 
direction of the axes is changed and the origin remains fixed ; 
these formulas express # and y as homogeneous functions of 
the first degree in #' and y'. If these values be substituted for 
# and y in the homogeneous polynomial of the second degree 


Ax? + 2 Bey + Cy’, 


the result will be a homogeneous polynomial of the second 
degree in a and y': 

Ala!” + a Bio'y' + @ pie 
In particular, the trinomial 

a? 4-2 ay cos 6+ 7° 

is transformed into 

a? +2 a'y' cos 0! + y”, 
6' being the angle between the new axes; because each of these 
trinomials gives the square of the distance of the origin from 
the same point of the plane. 

Consider the polynomial 


Ax? + 2 Bay + Cy? — 8 (a? +2 xy cos 6+ 4"), 
or (17) (A — 8) 2? +2(B— S cos 6) ay + (C— S)y’, 


174 PLANE GEOMETRY. BOOK III. 


in which the letter S designates an arbitrary constant; it will 
evidently furnish the transformed polynomial 


A'x"” + 2 B'z'y' + O44 is ie S (a* +. 2 a'y' cos @' as ve 
or (18) (A'—S) 2" +2(B'— 8 cos 6 a'y' + (C'— 8) y". 


One notices now that the values of S, for which one of 
the polynomials is the square of an integral function of the 
first degree in the variables which it involves, are the same. 
Assume, for example, that the first polynomial takes for a 
certain value of S the form (a#-+ by)’, a and b being con- 
stants; when a and y are replaced by their values (16), the 
function of the first degree, aw + by, changes into a function 
of the first degree a'x' + b'y', and the second polynomial takes 
the form (a'x'+ b'y')?, When the polynomials are squares, the 
equations which are found by equating their roots to zero 
represent the same straight line referred to the two systems of 
axes YOX, Y'OX'. 

The values of S, for which the polynomial (17) is a square, 
are the roots of the equation of the second degree 


(A — 8) (C—S) — (B— S cos 0)? = 0, 
or (19) S?sin?6— (A+ C—2Bcos6)S + AC— B’=0. 
The roots of this equation are represented by S, and S,; simi- 


larly, the values of S, for which the polynomial (18) is a 
square, are the roots of the equation 


(A! — 8) (C! — 8) — (B'— S cos 6? = 0, 
or (20) S?sin? 6! — (A! + OC! — 2 B' cos 6) S + A'C!— B" =0. 


The two equations (19) and (20) have the same roots, whence 
it follows ° 
(A+ C—2Bcos@_A'+C'—2B'cos6' 
sin? 6 ve Bin 0 : 
AC — B?_ A'C'— B® 


i sin?@  sin?6! 





(21) 4 











CHAP. III. REDUCTION OF THE EQUATION. 175 


Therefore, the two functions 
A+C—2Bcosd AC—B 


sin? 6 sin? 6 





of the coefficients of the equation of a conic and the angle 
between the axes preserve the same values when a transforma- 
tion of co-ordinates is made. 

A 
sin? 
suppose f different from zero; then, by transforming the origin 
of co-ordinates to the center of the conic, one has an equation 


whose constant term H has the value (§ 130) H =< : 





The quantity 5 possesses the same property. In fact, 


Since this constant term H remains the same whatever be 
the orientation and the angle between the axes and as an?@ 
sin 
remains constant when a change of axes is made, it follows that 


the same is true of 





Thus, the conic being referred to the 


axes xOy which include an angle @, if it be referred to new 
axes #'O'y' including an angle 6', and if the new equation be 
called 

A'a'? + yi B'e'y' + Cly” + a D'2!' — es E'y' + F' — 0, 
and the new value of A, A'= A'(C' F'— E”) +.---, one has 

Al A 
sin? 6’ sin? @ 

This relation will be satisfied if A', B', C', D', H', F', be 
replaced by their values as functions of A, B, C, D, E, F, which 
result from the formulas of transformation of co-ordinates. It 
still holds whatever A, B, C, D, H, F may be; that is, will be 
the same in the case where f is zero, although the argument 
used to establish this relation can no longer be applied to this 
case. 

The three quantities 


A+C-—2B 6 f A 
(23) ee 
sin? @ sin?@ sin? 


(22) 








are homogeneous with respect to the coefficients A, B, C, D, E, F; 
the first is of the first, the second of the second, the third of 


176 PLANE GEOMETRY. BOOK ILI. 


the third degree. If, therefore, the first member of the conic 
Aa? + 2 Bey + Cy? +2 De +2 Ey + F=0 


be multiplied by a constant A, that is, if A, B, C, D, HZ, F be 
replaced by A.A, AB, KC,---, the first of the three quantities 
(23) will be multiplied by A, the second by Av’, the third by 4°; 
whence it follows that a combination of quantities (23), homo- 
geneous and of the zero degree with respect to A, B, C, +--+, does 
not change if the factor 4 be introduced. Such, for example, 
are the two combinations 
f sin’ 0 A sin* 6 
(A+ C—2Bcos6)” (A+ C—2 Bcos 6)” 


found by dividing the second and third of quantities (23) by 
the square and cube of the first. One has then the two ex- 
pressions (24), which do not change when the axes are changed 
and all of the coefficients are multiplied or divided by the 
same factor. 

The condition A = 0 expresses that the conic is reduced to 
two straight lines; the condition f = 0, that it belongs to the 
genus parabola; the condition A+ C— 2 Bcos 6 = 0, that it is 

an equilateral hyperbola, that is, a hyperbola whose asymptotes 
are perpendicular. In fact, call m' and m" the angular coeffi- 
cients of the asymptotes ; the condition of perpendicularity is 


(25) 1+ (m'+m") cos 0+ m'm" = 0; 
further, these angular coefficients are roots of the, equation 
On? +2 Bm + A=), 


(24) 





which gives m'+m'!= — rik m'm" = 


values which, substituted in relation (25), give the required — 
condition. 


143. The magnitude of an ellipse or of a hyperbola depends 
on two numbers which are the lengths of the axes of the 
curves; the magnitude of a parabola depends on a single 
number, the parameter; finally, the magnitude of a conic 
reduced to two straight lines depends on a number which is 


CHAP. III REDUCTION OF THE EQUATION. 177 


their angle of intersection if they intersect, and the perpendicu- 
lar distance between them if they be parallel. It is next pro- 
posed to calculate these different quantities. 


Let Aa? + 2 Bay + Gy? +2 De +2Ey+F=0 


be the equation of a curve of the second degree referred to. axes 
inclosing an angle 6. If one put 


«-A+C—2Bcos6, f=AC—B, A=A(CF-F)+--, 
the three quantities 


f A 
1 € 
@) | sin?@ sin?@’ sin?6 








possess this property that any combination of these three quan- 
tities, homogeneous and of the degree zero with respect to the 
coefficients .A, B, C,---, has a constant value when "the co-ordi- 
nate axes are cay and the coefficients of the equation of 
the curve are multiplied or divided by the same factor, as has 
already been demonstrated. 

Assume, now, that the curve be an ellipse or a | ty perbala: 
on referring the curve to its center and its axes, its equation 
may be written 

ax’? + By’ — «8B = 0, 
where, in the case of a real ellipse, « = b’, B= a’; in that of a 
hyperbola, « = 0, B=—a’; in that of an imaginary ellipse, 
a=——bv’?,B=—a’. It is said in these three cases that @ and 
B are the squares of the lengths of the axes. The two com- 
binations 
eA A?’sin?6 


v #2. 


of quantities (1) being homogeneous and of the degree zero with 
respect to the coefficients, will have the same constant value if 
they be constructed for the reduced equation. In case of the 
reduced equation will . 


O=5 ex=a+p, f=a8, A=— ap’; 


therefore =—(« +); Ante = wf, 


178 PLANE GEOMETRY. BOOK III. 


and « and B are roots of the equation of the second degree, 
A A? n2 
(3) Tey yee eee LS 


which is called the equation of the squares of the axes. From 
the nature of the problem, this equation should have real roots ; 
this is easy t ify, b a 4 A? sin’6 

y to verify, because the quantity —{- -——} may 
then be written f 





2 
a (e- Af sin’) =24(A ~ Cy*sin?6 +[(A + C) cosd — 2 BY, 


which is necessarily a positive quantity. The roots are equal 


when 
We Cy 4 cong. 


the curve is then a circle. The roots will be equal and of 
contrary signs when «= 0; the curve is then an equilateral 
hyperbola. 

Suppose now that the curve of the second degree be a parab- 
ola; by referring it to its axis, and to a tangent at its vertex, 
its equation will take the form 


y? — 2px = 0. 


What is the value of the parameter p? The second of the 
three quantities (1) is zero. A homogeneous combination of 
the degree zero with respect to the coefficients can be formed 
from the other expressions in (1) by dividing the last by the 
cube of the first, which gives 

A sin‘ 6 


ee 





This last quantity constructed for the reduced equation is 
— p’; the equation which gives p is therefore 


A sin‘ 6 P 
JS 





144. In case the conic is reduced to a system of two straight 
lines which intersect, its equation can be written in the form 


y + ma’ = 0; 


CHAP. Ill. REDUCTION OF THE EQUATION. 179 


then A = 0, and one has, on forming the expression aL for 
f sin’6 
the reduced equation, 
os). 
? 


fsin? 6 8 ™ 





whence may be deduced two values of m which are reciprocals. 
If f be negative, the two straight lines are real and the values 
of m are negative: call ¢ the angle formed by the two straight 
lines, then will 

Pe 
f sin’ 0 





m= — tan? and — = 4 cot? , 


an equation which determines ¢. 


Finally, suppose that the curve is reduced to two parallel 
straight lines, and calculate the distance between them. In 
this case, two of the three quantities (1), f and A, are zero; the 
preceding combinations of the three quantities can no longer 
be employed. This case will be considered as a limiting case 
of the case when the curves have a unique center, and this in 
the following fashion. Let 


Ax? + 2 Bay + Cy? +2 Dv +2 Ey+ F=0 

be a conic reduced to two parallel straight lines; since A and 
C cannot be zero at the same time, on account of the condition 
f = 0, suppose that C differs from zero, and consider the auxil- 
lary curve 

(A +dA)a? + 2 Bay + Cy + 2 Dx + 2Ey+F=), 
which involves the parameter A. In this curve will 

«=A+C+A—2Bcos0, f= (A+A)C—B, 


A, =(A+A) (CF-—E’) +>. 
The expressions A, and f, reduce for A = 0 to A and f, that is, 
to zero. When A is different from zero, the auxiliary curve 
has a definite center; its equation reduces to the form 
ox? + By” — a8, = 9, 
wheré « and f, are roots of equation (3), which is written 


Ay 


2 
ie ae es i) sin?@ = 0. 
f, f, 


180 PLANE GEOMETRY. BOOK IT. 


When A approaches zero, f, and A, approach zero, and their 


CF — Ee 


ratio approaches the limit Therefore one of the 


1 
roots 6, of the preceding equation increases indefinitely, and 


the other approaches the limit 


es 
eC 


Further, the reduced equation may be written under the form 


a= sin? 6. 


hy 2 2 
—& + y? — % = 0, 


Bi 


which becomes 7? — « = 0, and the value found for « is the 
square of half of the distance between the two parallel straight 
lines to which the auxiliary conic is reduced for A = 0. 


EXAMPLES. 


iB 27° —38ay+3y+e—Ty+1=0. 

The curve is an ellipse, since the quantity AC — B? is posi- 
tive. In order to obtain the co-ordinates of the center, equate 
to zero the two partial derivatives 

4ea—38y+1=0, —32+6y—7T=0), 
whence Sy 4, A = — 13. 
If, keeping the axes parallel to themselves, the origin be 
transferred to the center C (Fig. 82), the equation becomes 
2a —3aHy,+3y/ —12 =0. 


Rotate now the axes through the angle @ given by the 


formula 
2B 


one 

The equation solved by the tables gives 
2 = 71° 33' 54", or a = 35° 46' 57", 

The angle « can also be found by a graphical construction; : 


lay off on the axes of w and y, beginning at the origin C, two 
lengths respectively equal to 1 and 3; the diagonal of the 


tan 2a= 58 


CHAP. Ill REDUCTION OF THE EQUATION. 181 


rectangle constructed on these two lengths makes with the axis 
of z an angle whose tangent is 3; the axis CX' is therefore 
the bisector of this angle. Whence A’ and C’ are obtained by 
the formulas 

; Al+ C'=5, A'— C!=—vi10, 


since B is negative. One has then 
S 10 i 10 
y (eae ae! ora? tvi0 
and the equation of the curve becomes 
(5 —V10)a? +(5 +V10)y? = 28; 


the intercepts of the curve on the axes are 


26 26 
CA = = ee =| = —* 
3(5 —+/10) 3(5 +-+/10) 


Lt. 22? —B5ay+5y—1=0. 














‘The curve is a hyper- eee 
bola (Fig. 84). The co-or- 
dinates of the center which 
are given by the equations — 
4a—5y=0, —5a+5=0 


Ds 
alll 
are 2 4. y= F, 














C Xi. 
whence H=1. i  N, 
: ple ere ae ae 

By transferring the origin VA ~ (oe 
to the center, the equation vA 
becomes 

20,7°—5ay,+1=0. Fig. 84.. 
The angle @ is given by the formula tan2a—— $, and one 


has A'+ C!=2, A'—C'4£—V/29; 


_2-V29 oy _2+V29, 
ya 2 
The equation of the curve referred to its axes 1s 


(2 —V29) @ + (2 +-V29) y2 +2 =0. 


whence A! 


182 PLANE GEOMETRY. BOOK III. 


The primitive equation does not contain a term in y*% One 
of the asymptotes is parallel to the axis OY. 


III. 4a? —12a0y +9? — 36" +100 =0. 


The curve is a parabola (Fig. 83). The terms of the second 
degree form a perfect square, and the equation can be written 


9(y — 22x)? — 3862+ 100 =0. 


Rotate the axes through an angle @ given by the formula 





tang =—2=0=5 whence a= 38° 41! 25", one will have 
C=_15 
; chee sie 
V13 
Dea o4 Ba 28, 
Ey VIB 


The equation of the curve referred to the axes OX, and OY, 


is therefore 

108 4 72 

pa | eae 
V1i3-— V1 

The co-ordinates of the vertex are found by combining with 

this equation the following 


72 
1 V/13 ’ 


36 pent Bae 
= = om : 
13-13 27.13-V13 
If the origin be transferred to this point, the equation becomes 
108 
a 
ay, 








13 yy Y. t+ 100 = 0. 


whence one finds 


n= 








13 y? — = 0. 


CHAP. Iv.  .CONCERNING THE ELLIPSE. 183 


CHAPTER IV 


THE ELLIPSE. 


145. It is proposed to construct the curve represented by 

the equation 
A'? + Cy+ H=0, 
in which the coefficients A’ and C' have the same sign. 

When the constant H is zero, the equation, being satisfied 
by «=0, y=0, represents a single point, the origin of co- 
ordinates. 

If the coefficients A’ and C' have the same sign as H, the 
equation cannot be satisfied by real values of and y, and does 
not represent a geometrical locus. 

Consider finally the case where these two coefficients have 
signs contrary to that of H, and put 





H jp H 
ap Ge 

the equation becomes 
(1) Jer ie) 

a- 
On solving it with respect to y, one gets 
(2) y=seVe—a, 
The ordinate y is real so long ¥ 
as the values of « are comprised = 7-— z ikl 





between —a and +a, and the eo 7 
same is true of w so long as the ee : 
values of y .are comprised be- 7 44 1 0 PL As 
tween —b and +0; if, there- 5 

fore, starting from the origin, one ,l* cade 
lay off on the a-axis to the right 
and ‘left two lengths OA, OA' Fig. 85. 




















184 PLANE GEOMETRY. BOOK III. 


equal to a, and on the y-axis two lengths OB, OB' equal to 3b, 
the curve is situated wholly within the rectangle CDEF con- 
structed on the two straight lines 4A', BB! (Fig. 85). 

As & increases from 0 to a, y decreases in absolute value 
from 6 to 0, which, on account of the double sign, furnishes 
the two equal arcs BMA, B'M'A. The same is true when x 
varies from 0 to — a, which giyes the two equal arcs BM,A', 
B'NA', equal to the preceding. These four equal ares form 
the ellipse. 


146. The straight line A'A is the axis of the ellipse, because 
to each abscissa OP correspond two ordinates Pe Pe, 
equal and of contrary signs. The straight line BB’ is also an 
axis of the ellipse; because, if the equation be solved with 
respect to x, one can verify in a similar manner that to each 
ordinate OQ correspond two abscissas QM, QM, equal and of 
contrary signs. The points A, A’, B, B', where the axes inter- 
sect the ellipse, are the vertices of the ellipse. The lengths 
A'A, B'B of the two axes are respectively equal to 2a 
and 2 0. 

The ellipse becomes a circle when the axes are equal. 

It is easy to see that the origin O is the center of the ellipse; 
in fact, let 2, y be the co-ordinates of any point M of the 
ellipse; it is evident that equation (1) is also satisfied by the 
values — x, —y; there is, consequently, a second point WN of 
the ellipse which has the co-ordinates — OP', — P'N respec- 
tively equal to the co-ordinates OP, PM of the point M, but 
measured in opposite directions; the triangles OPM, OP'N 
are equal; therefore OM= ON, and the line MON is straight 
because the angles POM, P'ON are equal. Thus the points 
M and N of the ellipse are two by two symmetrical with 
respect to the point O; therefore the point O is the center of 
the ellipse. 


147. In order to study how the distance from the center to 
different points of the ellipse varies, or the radius vector of 
the ellipse, find the equation of the ellipse in polar co-ordi- 


CHAP. IV. CONCERNING THE ELLIPSE. 185 


nates, when the center O is taken as the pole and the axis OA 
of the curve as the polar axis. If in equation (1) # and y be 
replaced by p cos w and p sin w, one has 





ag? m2 
(3) a wo , sin’ 


Pte (he b? 


Suppose a > b and write the equation in the form 


If w vary from 0 to = the ae ae ; increases, and, conse- 


quently, p decreases continually from : to b. The maximum 
value of p is a, the. minimum is 6, 


148. Represent sby x and y the co-ordinates of any point 
whatever of the plane and consider the polynomial 


The polynomial is equal to zero for a point situated on the 
ellipse (Fig. 86). Imagine that a mov- 
able point P starts from the point > 

M and moves along the prolongation 

of the radius vector OM: the two ae | 
co-ordinates # and y increasing in ab- 5 r 
solute value, the polynomial must in- eg 
crease indefinitely ; it takes, therefore, 

greater and greater positive values. Fig. 86. 

On the contrary, if the movable point 

travels toward the center, the polynomial diminishes and takes 
negative values. Thus, the polynomial 








is negative for every point situated within the ellipse, zero for 
points on the ellipse, and positive for every point situated 
without the ellipse. 


186 PLANE GEOMETRY. BOOK III. 


149. The squares of the ordinates perpendicular to an axis of 
the ellipse are proportional to the products of the corresponding 
segments formed on this axis. 

In fact, if # and y designate the co-ordinates of any point M 
on the ellipse (Fig. 85), one has, on account of equation (2) 

yp? b? yp b2 


—_*—__ = —, or ==. 
e—2 a (a—2z)(a+2) a 





But the two segments AP, A'P of the axis AA! are equal 
respectively to a —a and a+; one has, therefore, 
oe 8 seen es 
APX AP a 





Hence the square of the ordinate is to the product of the seg- 
ments formed on the axis in a constant ratio. 


150. The ordinates perpendicular to the major axis of an ellipse 
are to the corresponding ordinates of the circle constructed on this 
axis as a diameter in the constant ratio of the minor to the major 
axis. 

Let AA!’ be the major axis of the ellipse (Fig. 87); on this 
major axis as a diameter construct a circle; to the ordinate 
MP of the ellipse corresponds the ordinate M,P of the circle. 
Equation (2) may be written 


occas J tha 
Va? — 2? 


but Va?— 2 represents the ordinate M,P of the circle; one 
has, therefore, 


) x 


Rio 


MP _}b. 


MP a 





The minor axis enjoys the same property; the ordinate MQ, 
perpendicular to the minor axis, is to the corresponding ordinate 
M,Q of the circle constructed on this axis as a diameter in the 
constant ratio of the major to the minor axis. 

The ellipse is the orthogonal projection of a circle. Imagine 
that the circle AB,A' be revolved about the axis 4A’ through 


CHAP. IV. CONCERNING THE ELLIPSE. 187 


an angle ¢, such that cos ¢ = a the ordinate PM, of the circle 
a | 


will revolve about the point P, always remaining perpendicular 
to the axis 4A’; in this position MP will be the projection of 
M,P. In order to get the length of the projection, it suffices 


to multiply the length PM, by cos ¢, or by a which gives the 
a 


ordinate PM of the ellipse. Thus the projection of the point 
M, of the circle is the point M of the ellipse. Each point of 
the circle projecting thus into the corresponding point of the 
ellipse, it follows that the ellipse is the projection of the 
circle. 

One can also consider the circle as the orthogonal projection 
of an ellipse. Imagine the ellipse to be revolved about the 


, : Nee j 
axis BB' through an angle ¢@ whose cosine 1s —, the ordinate 
A 


QM of the ellipse will have for its projection the ordinate QM, 
of the circle described on BB! as a diameter, and the small 
circle will be the projection of the ellipse. 


151. The construction of the ellipse by points. From what 
precedes may be deduced a very simple method for construct- 
ing the ellipse by points. Construct on each of the axes of the 
ellipse, as diameters, a circle (Fig. 
87); draw from the center an arbi- 
trary secant intersecting the two 
circles in M,, M,; draw through 
the point M, a line parallel to the 
minor axis; through the point M, 
a line parallel to the major axis. 
The point of intersection M of 
these two lines belongs to the 
ellipse. After having determined 
in this manner a sufficient number 
of points, one connects them by a continuous line, and the 
ellipse is thus constructed. 

















152. Construct the points of intersection of an ellipse and a 
straight line. It is useful to be able to construct the points in 


188 PLANE GEOMETRY. BOOK III. 


which a given straight line MM" intersects an ellipse defined 
by its two axes AA’, BB! (Fig. 88) without tracing the ellipse. 
Thus, as has been seen, the ellipse can be considered as the 
orthogonal projection of the circle AB,A', described on the 
major axis 4A! as a diameter, the circle being revolved about 
AA' through an angle ¢ whose 
cosine is “ . Find in the plane 
of the circle the straight line 
M,M,', whose projection in the 
plane of the ellipse is MM’; 
let WV be any point of the line 
MM'; prolong the straight line 
BN till it intersects the axis 
AA!'in H; the line B,/7/ is pro- 
jected upon BH; consequently the point M,, where the line 
B,H intersects the ordinate QN, is projected into N. Simi- 
larly, any other point of the line 4/,M,' could be found; but it 
is more simple to begin with the point S, where the line MM' — 
intersects the axis; the line SN, has the given line in the 
plane of the ellipse as its projection. This line SN, cuts the 
circle in two points M4, ™,'; the ordinates MP, M,'P' will 
determine in the given line the two points M, M' where this 
line intersects the ellipse. 











Fig. 88. 


TANGENTS. ' 


153. The equation of the tangent to a curve of the second 


degree has already been found (§ 125); when the equation 
of the ellipse is put under the simple form 


a 


ar 


2 
+ —1=0, 
the equation of the tangent at the point M, whose co-ordinates 
are 2 and y, becomes 
4 
(4) are 
ba 


The angular coefficient of the tangent has the value ae 


ee ea sy 





CHAP. IV. CONCERNING THE ELLIPSE. 189 


One sees that at the vertices A and A’ the tangent is perpen- 
dicular to the axis A'A, that at B and B' it is parallel, and 
that, as the point of contact moves along the ellipse from A to 
B, the tangent makes with the axis A'A an obtuse angle, which 
increases from 5 to 7. 


'The normal, being perpendicular to the tangent, has the 
equation 





5 gece = 7 (X—2). 


154. The construction of the tangent at a point of the ellipse. - 
If in the equation of the tangent one put Y= 0, one obtains 


2 
the abscissa X =“ of the point 
wv 


T where the tangent intersects 
the prolongation of the major 
axis (Fig. 89). Since this value 
of OT is independent of the 4 
minor axis 26 and of the ordinate 
y of the point of contact, it fol- 
lows, that if several ellipses be 
constructed on the axis AA’, the Fig. 89. 
tangents at the points which have the same abscissas pass 
through the same point 7’ situated on the prolongation of the 
axis A'd. Among these ellipses is the circle 4B,A'; to con- 
struct the tangent to the ellipse at the point 1, draw a tangent 
to the circle at the point MM, situated on the same ordinate ; 
join the point M@ with the point 7, where the tangent to the 
circle intersects the prolongation of the axis A'A; the straight 
line MT, thus constructed, is the tangent to the ellipse. 
This construction is equivalent to regarding the tangent to 
the ellipse at the point Mas the projection of the tangent 
to the circle at the corresponding point M,. In fact, when the 
plane of the circle is made to revolve about the axis 4A! 
through an angle ¢, the point 7, where the tangent 7 meets 
the axis, remains fixed; the point M, projects into M, the line 
M,T has for its projection MT; it is the tangent to the 
ellipse. 














190 PLANE GEOMETRY. BOOK III. 


155. To draw a tangent through an exterior point P. Let x 
and y be the co-ordinates of the point P (Fig. 90). The equation 
of the chord of contact M'M has 
been found (§ 126). The determina- 
tion of the points of contact depends 
therefore on the solution of the two 
simultaneous equations 





@) 3+4=1, © += 


Fig. 90. 


By eliminating y, one gets the equation of the second degree 


x (x Lu 
ae +i) me 1 ht <0, 


of which the roots are the abscissas of the points of contact M 
and M' of the two tangents drawn from the point P. This 


equation, in which “ can be peeeiued a the unknown, will 
a 


have real roots if the condition a a > 0 be satisfied ; 
that is, if the point P be Sen the ellipse. 











oe 





Fig. 91. 


It is easy to construct geometrically the tangents drawn 
from the point P, by regarding the ellipse as the projection 
of the circle AB,A' (Fig. 91). Seek in the plane of the circle 
the point P,, whose projection in the plane of the ellipse is 
the point P. Draw in the plane of the ellipse the straight 


CHAP. IV. CONCERNING THE ELLIPSE. 191 


line PB, which is prolonged till it intersects the axis in H; 
the straight line HB,, having HB for its projection, will pass 
through the point P, and determine this point. Draw from 
the point P, to the circle the tangents P.M, P,M,', which one 
prolongs till they intersect the axis in J’ and 7"; the straight 
lines PT, PT', projections of the tangents to the circle, will be 
tangents to the ellipse, and the points of contact M and M' 
will be situated on the ordinates of the points Mj, My’. In 
order that these constructions be accomplished, it is not neces- 
sary that the ellipse be drawn. 


156. To draw a tangent parallel to a given straight line. Let 
y= mex be the equation of the given straight line OL, which 
may be supposed to be 








drawn through the center le 
Fig. 92). Call a and CTF SX 
8 y V L 4 WA w 
a << 1, / sy Re 
the unknown co-ordinates YN ff WN. 
bg } Wee SS y \ =< 
of the point of contact M; _z’ PSA N 
i : : S41 A a) A 
this point being on theel- “S\\ i [pica 
. ° PNG. i é : 
lipse, one has the equation NLA 4 
; / Zw 
iy i y? 28 1 HM; San cae oe 
nape ; 
Fig. 92. 


the angular coefficient of the tangent being equal to m, one 
has a second equation 


These two simultaneous equations determine the two un- 
known quantities «and y; the first represents the given ellipse ; 
the second a straight line passing through the center; the 
points where this straight line meets the ellipse are the points 
of contact. 

It is easy to construct these tangents geometrically. Deter- 
mine first in the plane of the circle the diameter OZ,, whose 
projection in the plane of the ellipse is OL; it is sufficient to 
join the point B with any point Z of the line OZ, and prolong 
the line BZ till it intersects the axis in H; then draw BH 





192 PLANE GEOMETRY. BOOK III. 


and locate the point of intersection of this line with the ordi- 
nate of the point Z; the point Z being the projection of the 
point Z,, the line OL is the projection of OLZ,. One draws to 
the circle the tangents MT, M',T"', parallel to OZ,, and through 
the points 7’ and 7", where three tangents intersect the axis, 
the lines 7'M, T'M' parallel to the line OZ. One has the tan- 
gents required; because the projections OL, 7'M of the parallel 
straight lines OZ,, TM, are also parallel. The points of con- 
tact M and M’ are determined by the ordinates of the points 
M, and M',. 


157. The equation of a tangent to the ellipse may be found 
in other forms which it will be useful to know. 

If one designate by « the angle which the perpendicular let 
fall from the center to the tangent makes with the axis of x 
and by p the length of this perpendicular, the tangent will be 
represented by the equation (§ 83) 


Xcosa+ Ysne—p=0; 
or, comparing it with equation (4), one has the relations 


x y 
a d fs a 


acosa bsine p WVa'cos?a+ b’sin?«’ 














whence p= Va’ cos’ a + b* sin? «. ef 


Then the tangent will have the equation 





(6) Xeosa + Ysine = Va?cos’?« + b?sin?a. 


The equation of the tangent may also be found by seeking 
the points.of intersection of an ellipse and of a straight line, 
and then expressing the condition that these two points should 
coincide, as has been done in case of the circle (§ 94). One 
obtains in this way the equation of the tangent in the form 


(7) y= metVa' m + 6’ 


CHAP. IV. CONCERNING THE ELLIPSE. 193 


158. As an application, it is proposed to find the locus of 
the vertex of a right angle which e 
circumscribes the ellipse. Sup- > 
pose tnat one draw through an ee a 
exterior point P (Fig. 93), whose ee 
co-ordinates are w' and y', tangents 
to the ellipse; on account of the 


tangent passing through the point : 
P, one will have the equation of 


condition <a 
y! = ma! + Ven 


Fig. 93. 

















in which the angular coefficient m 
isunknown. ‘This equation, written in an integral form, 
(ar an!) m? +2 2e'y'm + (0? a y”) = 0, 

is of the second degree; its two roots determine the directions 
of the two tangents drawn from the point P to the ellipse, 
and, consequently, determine these tangents. The two tan- 
gents drawn from the point P will be rectangular if the prod- 
uct of the two values of m be equal to — 1, which will be the 
case if the co-ordinates of the point P satisfy the relation 


b? — y” 
a io; —1, or x? + y? =a? + D. 


Hence the locus of the vertex of a right triangle circumscribed 
about an ellipse is the circle circumscribed about a rectangle 
constructed on the axes. 


DIAMETERS. 


159. The general equation of a diameter of a curve of the 





second degree has been found in Y 
§ 131. On representing by . < D 

cs (a, Y) =0 a, ~ 
the equation of the curve, and by m “4 0 ue x 
the angular coefficient of the chords Zs 
parallel to MM' (Fig. 94), one has fre 





seen that the equation of the diameter Fig. 94. 
N 


194 ‘PLANE GEOMETRY.  - BOOK UL. 


DD! may be written in the form f!, + mf!,=0. The equation 
of the ellipse being referred to its axes, the equation of the 
diameter reduces to 





On representing by m’ the angular coefficient of the diameter 
DD", one has, between the direction of the chord and that of 
the diameter, the relation 


2 


(8) mm! = -<. 
It has also been shown that if the chord MM" be drawn par- 
allel to the diameter DD', the diameter OF, which bisects this 
chord, has the angular coefficient m; the two diameters DD’, 
EE' form a system of conjugate diameters, and their angular 
coefficients m' and m are connected by relation (8). 

This relation shows that the two angular coefficients m and 
m' have opposite signs, and, consequently, that the two semi- 
conjugate diameters OD and OE, situated on the same side 
of the major axis, are situated on opposite sides of the 
minor axis. If the first start from OA and revolve from 
OA toward OB, the second starts from OB and revolves 
toward OA’. 


160. The tangent at any point D of the ellipse is parallel 
to the diameter HE’, the conjugate of the diameter DD’, which 
passes through the point of contact. In fact, if one call # and 
y the co-ordinate of the point D, the diameter OD has the 


angular coefficient m = /. the coefficient of the tangent at the 
point D is m'= — ay? these two coefficients satisfy the rela- 
2 
tion mm! = — we 
a 


This property may be described more clearly by imagining 
that the secant MM', moving parallel to the diameter ELE’, 
recedes continually from the center; the two points of inter- 
section M and M' approach more and more the middle of the 


CHAP. IV. - CONCERNING THE ELLIPSE. 195 


chord, and end by coinciding with D; then the secant becomes 
a tangent at D. 


161. The properties of conjugate diameters are exhibited at 
once on considering the ellipse as the projection of the circle. 
Two rectangular diameters OD, OF, 





(Fig. 95), in the plane of the circle, Ei 7 

form a system of conjugate diameters, B: Ps 
because each of them bisects the chords | DI 
parallel to the other; the parallel chords 41-43 Pf 


are projected into parallel chords in the 
plane of the ellipse; the mid-point of 
the chord has for its projection the mid- 
point of the projection of the chord; 
each of the diameters OD, OF, the prajcetions of the diame- 
ters OD,, OE), bisects therefore the chord parallel to the other ; 
they are therefore conjugate diameters of the ellipse. It is 
easy to deduce the relation which exists between the angular 
coefficients m and m!' of the two conjugate diameters. If m, 
and mz, be called the angular coefficients of the two conjugate 





Fig. 95. 


: ; b b 
diameters OD,, OF, of the circle, one has m = —m,, m!=-m',; 
a a 
b? ; ‘ are 
whence mm'=-—mym',; since the conjugate diameters of the 
a 


circle are perpendicular, one has mym';=—1; it follows then 
that mm! = — Y 

Being given OD, one can find its conjugate OF, without 
drawing the ellipse. One constructs the diameter OD, whose 
projection is OD; and draws the diameter OF, perpendicular 
to OD,, and ae OE,; the projection ai will be the 
diameter required. 


162. The ellipse referred to two conjugate diameters. . Owing 
to what has been said in § 136, when two conjugate diam- 
eters OD, OF (Fig. 96) are taken as axes of co-ordinates, the 
equation of the ellipse can be written 


Alte’ + Oy? + H=0. 


196 PLANE GEOMETRY. BOOK III. © 


Since the coeffieients A" and C" have the same sign, contrary to 
that of H, if one put 


Y’ 
K fs Eee H 
~ Na a” —_—_— Aer b =o qi 




















E B ee 

. < the equation takes the form 
B rr = a? y” 
D ES H @) a? 2 b” ae 
Z, 
Se which is the same form as 
oe that of the curve referred 
ee: to its axes. 


It follows that the calculations employed in demonstrating 
the properties of the ellipse, when the equation of the curve 
was referred to its axes, and in which the co-ordinates were 
not supposed orthogonal, could be repeated with the equation 
of the curve referred to a system of conjugate diameters. 
Thus, the ellipse being referred to a system of conjugate 
diameters OD and OE, the tangent will have the equation 


g'X! eA 
a" Ag oe 








However, the equation of the normal does not preserve the 
form which corresponds to the axes OA and OB. 


THEOREM OF APOLLONIUS. 


163. The theorem of Apollonius admits of an easy demon- 
stration by the method of §142. Imagine the ellipse referred 
successively to its two axes and to a system of conjugate 
diameters forming an angle 6. By the formulas of transfor- 
mation of co-ordinates, the binomial 


is transformed into oe be 


~ CHAP. IV. CONCERNING THE ELLIPSE. 197 
Similarly the binomial =a” - 
becomes al? + y+ 2a'y! cos 8, 


since each of the two expressions represents the square of 
the distance of the origin from the same point of the plane. 
Whence it follows that the polynomial 


ae | 
or 
dere | Eg te 
10 ae ae again 
( ) & e+ (a i) 


in which ) plays the réle of an arbitrary constant, is trans- 
formed into 


12 12 1 a 
a = es + y" + 22'y' cos 8), 


or 
oy os 6 jie 
SO coat) les wh da C9 


The values of A, which make one of the polynomials (10) or 
(11) a perfect square, being the same, the two equations 


See aes, 
oe AW ey Weed 


or 

(12) (A — a) (A — 0’) = 0, 

and ale 
aA eK ? f 

or 

(13) MM (a? +5") + ab” sin’ 6 = 0, 


have the same roots. It follows, therefore, that the two roots 
of equation (13) are equal respectively to a? and 6’, whence 
follow the two relations: 


(14) a”? + 6? = a? + 0; 


(15) a?b" sin? 6 = a7b*, or a'b' sind = ab. 


198 PLANE GEOMETRY. BOOK III. 


The preceding equations furnish the following theorems : 


1° The sum of the squares of any two conjugate diameters of 
an ellipse is constant and equal to the sum of the squares of the 
axes. 


2° The area of the parallelogram constructed on two conju- 
gate diameters is constant and equal to that of the rectangle con- 
structed on the axes. 

Relations (21) of § 142 give immediately the two equations 
(14) and (15). 


164. These theorems may easily be demonstrated by con- 
sidering the ellipse as the projection of a circle. 

Two conjugate diameters OD, OE of the ellipse are the pro- 
jections of two perpendicular diameters OD,, OE, of the circle 
(Fig. 95). The angles D,OP, E,OQ being complementary, the 
right triangles D,OP, E,OQ are equal, and one has 


O0Q=D,P; but OD, = OP’ + D,P’; 
it follows that OP’ + 0Q = a2. 


The lengths OP and OQ being the projections of the two semi- 
conjugate diameters OD and OF on the major axis of the 
ellipse, the sum of the squares of these two projections is 
constant, and one has 
a” cos? a + b” cos? B = a’, 
on representing by « and 8 the angles which the semi- diameters 
OD and OF make with the axis OA. 
Similarly for the other axis, one has the projection of the 
two semi-conjugate diameters on the minor axis equal to the 
ordinates DP and EQ. DP =" DP, EQ =" EQ, and, 


consequently, 





ees bh? 9 ‘ 
DP’ + EQ = (DP + EQ). 
The lengths #,Q and OP being equal, one has 
D,P* + E,Q = D,P* + OP’ =a’, 


CHAP. IV. CONCERNING THE ELLIPSE. 199 
and, consequently, DP’ + EQ =’, 
or a” sin? + b? sin? B = b*. 


On adding member to member the two preceding relations, 


one obtains 
a” + b? oes a + b?, 


165. In order to demonstrate the property respecting the 
area of the parallelogram, one makes use of the following 
theorem : 


The projection of a plane area upon any plane is equal to the 
projected area multiplied by the cosine of the angle between the 
planes. 

For this purpose consider a triangle ABC 
(Fig. 97) having an edge AB parallel to the 
plane of projection; one can assume that 
the plane of projection passes through this 
edge AB; from the vertex C, drop upon this 
plane a perpendicular CC’, and, in this plane, 
draw C'D perpendicular to AB; the straight 
line CD will also be perpendicular to 4B 
and the angle CDC' is the measure of the 
dihedral angle of the two planes. From the 
construction it follows that 


C'D = CD cos 4, 








AB-C'D _ AB- CD 


h 
whence 9 9 





cos ¢, 


and, consequently, 
AC'B = ACB x cos ¢. 


Thus the area of the triangle AC'B is equal to that of the 
triangle ACB multiplied by cos ¢. 

Suppose now that the triangle ABC (Fig. 98) has no side — 
parallel to the plane of projection; this plane can be passed 
through a vertex A, in such a way that the other two vertices 
may lie on the same side; the plane of the triangle produced 


200 PLANE GEOMETRY. BOOK III. 


intersects the plane of projection in a straight line AT, and the 
line CB intersects this plane in the point 

o I; the triangles AIC, AJB project into 
AIC", AIB', and one has, after what has 
just been proven, 


AIC" = AIC cos ¢, 





4 AIB' = AIB cos ¢, 


whence, by subtraction, 





Fig. 98. AB'C' = ABC cos ¢. 


The theorem, being demonstrated for a triangle, may be 
extended toa plane polygon, since it can always be decomposed 
into triangles, and similarly to a plane area bounded by any 
closed curve; because this plane area may be regarded as the 
limit of the area of an inscribed polygon, of which the number 
of sides is increased indefinitely, in such a way that each 
approaches the limit zero. 

When the ellipse is regarded as the projection of a circle, the 
parallelogram constructed on the two conjugate diameters is 
the projection of a square circumscribed about the circle; the 
square having a constant area equal to 4a’, that of the et 
lelogram is also constant and equal to 4a? cos ¢, that is, to 4 ab. 


t 
v 


AREA OF THE ELLIPSE. 


166. The same theorem furnishes immediately the area of 
the ellipse. The ellipse being the projection of a circle, its 
area is equal to that of the circle za’, multiplied by cos ¢ or by 


iy which gives zab. 
a 


CHAP, IV. CONCERNING THE ELLIPSE. 201 


EQUAL CONJUGATE DIAMETERS. 


167. It has been noticed (§ 159) that the two semi-conju- 
gate diameters OD, OE lie on opposite 
sides of the minor axis OB (Fig. 99). ar G 
One knows that the radius vector of the oe 
ellipse increases in length as it is rotated , 
farther from the minor axis; in order 0 
that two conjugate diameters may become 
equal, it is therefore necessary that 


they make equal angles with the minor 
axis OB, which will take place when the angles « and £B are 
2 




















Fig. 99. 


supplementary. One has, therefore, tan?a = ee and, conse- 
a 
quently, tana = 2; hence the equal conjugate diameters OG 
a 


and OH coincide with the diagonals of the rectangle constructed 
on the axes. 
It follows from the relation a”? + 6”? = a? + b? that 


2 2 
a+b 
a’? = ’ 





2 
and the equation of the ellipse, referred to its equal conjugate 
diameters, is a? + B 


ety? = 





2 ] 
it has the same form as the equation of a circle, only the co- 
ordinates are oblique. 

This equation shows that the sum of the squares of the dis- 
tances of each of the points of the 
ellipse from two equal conjugate 
diameters is constant. In fact, a 
let 6 be the angle between the > 
two equal conjugate diameters; 4 ~ 
MP and MQ the co-ordinates of 
the point M (Fig. 100); MH and 
MF the perpendiculars dropped ; 
from M upon these Soeananatie Fig. 100. 
one has ME = y'sin 0, MF =-2'sin 6; whence 
29 (+ 6*)sin’@_ 2 a*b? 

2 2 b? 


























ME* + MF” = (@" + y”) sin 


202 PLANE GEOMETRY. BOOK III. 


SUPPLEMENTARY CHORDS. 


168. Two chords MC, MC' in an ellipse are called supple- 
mentary chords, if they be drawn from any point of the 
ellipse to the extremities of a diameter 


X. : CC! (Fig. 101). 
E L> pO supplementary chords are parallel 
to the corresponding conjugate diameters. 
c Draw, in fact, the diameters OD and OE 
parallel to the supplementary chords MC’, 
Fig. 101. MC. In the triangle CMC’, the two sides 


CC! and CM are divided by the line OD, 
parallel to C'M, into parts which are proportional; the center 
O being the mid-point of CC’, it follows that the diameter OD 
divides the chord C'M into two equal parts, and, consequently, 
every chord parallel to the diameter OE. Similarly, the diam- 
eter OE bisects the chord C'M, and, consequently, every chord 
parallel to OD. Therefore the two diameters OD, OE, parallel 
to the supplementary chords MC', M C, are conjugate. 

Conversely, if straight lines be drawn from the extremities 
of a diameter CC’ parallel to two conjugate diameters OD, OE, 
these straight lines intersect on the ellipse; draw C'M; the 
supplementary chords MC, M C' being parallel to the two con- 
jugate diameters, the second chord C'M will be parallel 
to OD. 


ft 
ve 


169. The study of the variation of the angle formed by two 
conjugate diameters is thus reduced to the study of the varia- 
tion of the angle formed by two sup- 
plementary chords, that is, of the 
angle inscribed in a semi-ellipse. In 
order to simplify the discussion, one 
draws the two supplementary chords 
through the extremities of the major 
axis (Fig. 102). The angle AMA', 
represented by 6, is equal to the dif- 
ference between the angles MAX, 








“Fig: 102. 


CHAP. IV. CONCERNING THE ELLIPSE. 203 


MA'X. Since the two straight lines 4M, A'M have the angu- 





lar coefficients —” , —4_, one has 
x—-a e+a 








\ siete 
_@—-a @+4 2 ay 
tan 6 = oa y py — a 
x? — a? 


and, by replacing 2? by its value deduced from the equation of 


the ellipse, 
2 ab? 


(a? — b*) y 


If the point M describe the upper portion of the ellipse ABA’, 
the tangent being negative, the angle is obtuse; when the point 
M is at the point A, that is, when y= 0, the angle is right ; 
the point M traveling from A toward B, y increases; the abso- 
lute value of tan @ diminishes; the obtuse angle ¢@ increases 
also, and acquires its maximum value at B; thus one has y = b 

2ab 
a? — b? 
and traces the elliptical quadrant BA’, the angle 6 diminishes 
from its maximum value to a right angle. 

Whence it follows that the angle between the semi-conjugate 
diameters OD, OF, situated on the same side of the major axis, 
is obtuse, and varies from a right angle to the maximum value 
ABA'; the conjugate diameters, which embrace the maximum 
angle, being respectively parallel to the supplementary chords 
A'B, AB, and, consequently, forming equal angles with the 
minor axis OB, are equal. 

The variation of the obtuse angle DOE of two conjugate 
diameters has been studied; the acute angle DOE!’ varies in 
an inverse manner. This angle is obtained directly by 
drawing the corresponding supplementary chords through the 
extremities of the minor axis BB’. When the point M 
describes the quadrant of the ellipse BA, the inscribed angle 
diminishes from a right angle to the minimum value BAB’, 
the supplement of the obtuse maximum value ABA’. 


tan é= — 





and tan @ = — When the point M passes the point B 


204 PLANE GEOMETRY. BOOK III. 


170. When the ellipse has been drawn, the center and the 

axes can be determined graphically. In order to find the 
center, one draws two parallel chords sufficiently distant from 
each other, and then joins the mid-points of these chords, 
which will determine a diameter, whose mid-point will be the 
center. If on this diameter a semi-circle be constructed, and 
the point where this semi-circle intersects the semi-ellipse be 
joined to the extremities of the diameter, one will have two 
supplementary chords which are perpendicular; the parallel 
diameters, forming a system of perpendicular conjugate diam- 
eters, will be the axes of the ellipse. 

In a similar manner, the two systems of conjugate diameters 
which include a given angle having as limits the minimum and 
maximum values, can be constructed; it will suffice to con- 
struct on a diameter a segment which will circumscribe an 

angle equal to the given angle. 


171. Being given two conjugate diameters, construct the corre- 
sponding ellipse. Let DD', EE' (Fig. 103) be the given conju- 
gate diameters, whose lengths are represented by 2 a' and 20!. 
The equation of the ellipse, referred to these two conjugate 
diameters, is 


Draw through the center the line Z,Z4, perpendicular to DD’, 
and take OZ,=OE; the ellipse which has the_axes DD’, 
E,E', referred to these axes, 
is represented by the same 
equation. Whence it follows 
that the ordinates MP, M,P, 
which correspond to the 
same abscissa OP, are equal 
to each other. Imagine that 
different points of the el- 
lipse DE, D', whose axes are 
known, are constructed by the process described in § 149; 
let M, be one of these points, M,P its ordinate; if one draw 
through the point P, PM parallel to OE and equal to P.M), one 











” Fig. 103. 


CHAP. IV. CONCERNING THE ELLIPSE. 205 


will have the point M of the ellipse required. Each point of 
the first ellipse will give a corresponding point of the second. 
The first ellipse is deformed into the second by revolving each 
ordinate PM, about its foot P through a constant angle. 

The same method of transformation can be applied to the 
tangent. The tangent at the point M is represented by the 
equation 


ex b a 
(4) =i a 1, 


written in oblique co-ordinates; this equation represents also 
the tangent at the point M, if written in rectangular co-ordi- 
nates. These two tangents intersect the prolongation of the 
diameter DD! at the same point 7, the abscissa of which is 
found by making Y = 0. 

Instead of constructing the ellipse by points, as has been 
explained, the axes of the ellipse can first be constructed, and 
then the ellipse itself by means of its axes. The determina- 
tion of the axes depends upon the following theorem : 


172. Any two conjugate diameters determine on a fixed 
tangent PQ two segments DP, DQ, whose product is constant 
and equal to the square of the 
semi-diameter OE parallel to the 
conjugate (Fig. 104). If one 
take as axes of co-ordinates ~_, 
the diameter OD, which passes 
through the point of contact 
and its conjugate OH,.and if 
one calls a’ and b!' the lengths 











of these semi-diameters, the Fig. 104. 
equation of the ellipse is 
ee 
aes 
Let y= mer, y= m2, 


be the equations of two conjugate diameters O.A, OB; according 
to the remark made in § 160, the angular coefficients will be 


206 PLANE GEOMETRY. BOOK III. 


12 
connected by the relation mm! = — ° If in these equations 


one put «= a', one finds DP = — ma', DQ = m'a'; whence 


DP. DQ=— mm'a” = b". 


173. This theorem may easily be demonstrated by consider- 
ing the ellipse as the projection of a 
circle. Let OA;, OB, (Fig. 105) be 
@: two perpendicular diameters of the 
Mi circle, P,Q, the tangent at any point 
M,; draw the radius OM, and the 
‘ i radius ON, parallel to the tangent; 
in the right triangle P,OQ,, one has 
way, MP, - MQ, = OM/ = ON/. 


Fig. 105. 


Ni 








When the figure is projected, the di- 
ameters OA,, OB, furnish two conjugate diameters of the 
ellipse, the tangent P,Q, a tangent to the ellipse, and the line 
ON, a parallel to this tangent; the lines M|P;, MQ, OM, 
being parallels, have the projections MP, MQ, ON, which are 
proportional to them; there also exist, therefore, between these 
projections the relation 


MP. MQ = ON? 


174, Suppose that the two conjugate diameters OA and OB 
be the axes of the ellipse (Fig. 104). The circle déscribed on 
PQ asa diameter passes through the point O, and the ordinate 
DH, perpendicular to PQ, is equal to OF. Whence follows a 
simple device for constructing the directions of the axes, when 
one knows the two conjugate diameters OD and OE. One 
draws through the point D a line parallel to OZ; this parallel 
will be tangent at the point D; on this line one erects a per- 
pendicular DH equal to OE, and describes a circle having its 
center on PQ and passing through the points O and H; the 
straight lines OP and OQ which connect the center with the 
two points P and Q, where the circle intersects the tangent, 
will give the direction of the axes. 


CHAP. IV. CONCERNING THE ELLIPSE. 207 


175. It remains to determine the magnitudes of the axes. 
From the relations 


a+b?=—a"?+b", ab=a'd' sind, 
established in § 163, one deduces 


(a — bP? =a? + 6° —2a'b'sind =a? + b" — 2a'd' eos (5 — a), 


(a+ bP? =a? +b" + 2a'b' sind =a” + 6” — 2a'b' cos G +. 0). 


Since one can suppose that 6 designates the angle included by 
the conjugate diameters, one sees from these formulas that 
a—bis the third side of a triangle of which the other twe 


sides are a' and 0! and the included angle = — 6. This triangle 


is the triangle ODH (Fig. 104); because the angle ODH is 
equal to . 6, and the two sides DO and DH are equal to a! 


and b'; thus the third side OH will be equal toa—b. Simi- 
larly a + b is the third side of the triangle of which the other 
two sides are a! and b!, and the included angle the supplement 
of the preceding; this triangle is the triangle ODK, which is 
obtained by prolonging the perpendicular DH till the prolon- 
gation is equal to itself; the third side OX will determine 
a+06. If about the point O as center, with OH as a radius, 
one describe a circle, the length AT will be equal to the major 
axis 2a, the length AZ to the minor axis 2b. 

One remarks that the major axis, which should: lie within 
the angle formed by the conjugate diameters, has the direction 
OA, the bisector of the angle HO, the minor axis is the 
bisector of the supplementary angle. 


EXERCISES. 


1. Find the locus of the vertices of the parallelograms 
constructed on the conjugate diameters of an ellipse. 

2. Find the locus of the mid-points of chords drawn through 
the same point in an ellipse. 


208 _ PLANE GEOMETRY. BOOK III. 


3. A chord of a circle moves parallel to itself; straight 
lines parallel to two given straight lines are drawn through 
the extremities; find the locus of the point of intersection of 
the parallels. 

4. Of all the parallelograms circumscribed about the same 
ellipse, the parallelograms constructed on two conjugate diam- 
eters have a minimum area. 

5. Of all the parallelograms inscribed in the same ellipse, 
those whose diagonals form a system of COMES Te diameters 
have a maximum area. 

6. Of all the ellipses inscribed in the same parallelogram, 
find the greatest. 

7. Find the smallest of all the ellipses circumscribed about 
the same parallelogram. 

8. Among all the systems of conjugate diameters of an 
ellipse, the axes form a minimum sum and the equal conjugate 
diameters a maximum sum. 

9. Inscribe in an ellipse a chord with a given direction 
such that the sum of its length and of the distance of its mid- 
point from the center be a maximum; find the locus of the 
mid-point of this chord when the direction varies. 

10. A straight line moves parallel to itself in the plane 
of two others; one takes on it a point such that the sum 
of the squares of its distances from the intersections with the 
fixed lines be constant; what is the locus described by the 
point ? 

11. Being given any two ellipses, one can detérmine two 
directions parallel at the same time to two conjugate diameters 
of each of the ellipses; pass a third ellipse of which the equal 
conjugate diameters are parallel to these two directions through 
the points common to the two curves. 

12. An ellipse revolves about its center; one draws tan- 
gents to the ellipse at the points in which it intersects a fixed 
straight line; find the locus of the point of intersection of 
these tangents. 

13. Being given a circle and a fixed straight line passing 
through its center; a movable straight line equal to the 
radius is supported by one of its extremities on the circum- 


CHAP. IV. CONCERNING THE ELLIPSE. 209 


ference, by the other on the line; find the locus of a point 
on the movable straight line. 
14. Find the area of the ellipse defined by the equation 


Ax? + 2 Bay + Cy? = 1. 


15. A triangle being inscribed in an ellipse, if one call R 
the radius of the circumscribed circle and d, d', d' the semi- 
‘diameters parallel to the sides, one has 


dd'd"' 

ab 

16. Any rectangle being circumscribed about an ellipse, the 
parallelogram whose vertices are the points of contact has a 
constant perimeter, and two consecutive sides make, with the 
tangent, equal angles. 

17. Beginning at any point on the ellipse, one lays off on 


R= 





2 
the normal a length equal to = k being a constant and p the 
P 


perpendicular dropped from the center upon the tangent; 
find the locus of the extremity of this line. 

18. Being given an ellipse and the circle constructed on its 
major axis or diameter, one draws normals to the circle and to 
the ellipse at points situated on the same perpendicular to the 
major axis; find the locus of the point of intersection of the 


normals. 
Oo 







210 THE HYPERBOLA. BOOK II. 


CHAPTER V 
THE HYPERBOLA. 


176. Cottstruct/the Jocus defined by the equation 


in which A! and C' have contrary signs. ( 


When the constant is zero, the equation, solved with 


respect to y, gives < a 
oa \ —? 


ae 
it represents two straight lines passing throu rigin. 
One gives the coefficient C' the same sign as H, and the 


coefficient .A' the opposite. If one put 


134 HT 
PY anes ae b? = a iy, | 
the equation becomes 
oy 
(1) ahr bi 


Solving the equation with respect to y, one has 


(2) yas Vie 2 
The ordinate 7 is real for values of x greater than a in absolute 
value. If, therefore, beginning at the origin, one lay off on 
the axis of z, to the right and the left, two lengths OA, OA' 
equal to a, and draw through the points A and A' lines parallel 
to the axis of y, no point of the curve will le between these 
parallels. 

When z increases from a to + , y increases from 0 to + 
in absolute value, which, on account of the double sign, fur- 
nishes two infinite ares AD, AD', symmetrical with respect 
to the axis of x. Similarly, when 2 varies from — a to —o, 
one gets two infinite ares A'E, A'E’, symmetrical with respect 


CHAP. V. CONCERNING THE HYPERBOLA. ati 


to the axis of y. These four equal arcs form the two branches 
of the hyperbola. 

The hyperbola has a center and 
two axes. The axis 44’ only in- 
tersects the curve; for this reason | 
it is called the real or transverse 
axis; the other axis does not 
meet the curve; one calls it the 
non-transverse or imaginary awis ; 
the length AA’ of the transverse 
axis is 2a; by analogy, the length 
of the non-transverse axis is Fig. 106. 
called 20, and on this axis one lays off OB and OB! equal in 
absolute length to &. The points A and A’ are the two vertices 
of the hyperbola. 














177. The squares of the ordinates perpendicular to the trans- 
verse axis are proportional to the products of the corresponding 
segments on this awis. 


In fact, from equation (1) it follows that 








ee ee y ee 

ea a” (a+a)(e—a) a? 
therefore MP" = bt 
APx< AP ¢ 


178. Asymprotres.—It has been found (§ 130) that, when 
the origin of co-ordinates coincides with the center of the hyper- 
bola, the equation of the asymptotes is found by suppressing 
the constant term in the equation of the curve. The two asymp- 
totes RA', SS' will have in this case the equations 
(3) aay or veg oe, 

a 

It can be easily verified that the difference MN of the ordi- 
nates of the straight line OR and the are AD, has the limit 
zero; because this difference can be expressed by 


(ae aan 2+ ea 





oi? PLANE GEOMETRY. BOOK ITI. 


The are AD lies wholly within the angle ROX and ap- 
proaches indefinitely the line OR, which is its asymptote. The 
lines OR', OS, OS' are for a similar reason the asymptotes 
of the ares A'EH', A'E, AD'. According to equation (8), the 
asymptotes ’R, S'S are the diagonals of the rectangle con- 
structed on the axes. 


179. ConsuGATE Hyprrsoias.—Two hyperbolas are said to 
be conjugate, when they have the 
same center and the same axes, 
the real axis of the one being 
the imaginary of the other. 
Thus the proposed hyperbola has 
as conjugate another hyperbola 
whose transverse axis is 2b and 
imaginary axis 2a (Fig. 107). 
The equation of this second hy- 
perbola is 














(4) ae A 


Two conjugate hyperbolas have the same asymptotes, since 
the rectangle constructed on the axes is the same for both 
curves. One of the curves lies wholly within the vertical 
angles ROS', R'OS, the second within the other vertical angles 
ROS, B'OS". 


180. THe EquinATERAL Hypersoras.—A hyperbola is said 
to be equilateral when the axes 2a and 26 have the same length. 
In this case, the rectangle of the axes becomes a square, and 
the asymptotes are perpendicular to each other; the conjugate 
hyperbola is equal to the first; for the two curves will coincide 
when one revolves the latter through a right angle about its 
center. 

The condition that the general equations of the second degree 
represents an equilateral hyperbola, has been previously given 
(§ 144); this condition is A + C—2 Bcos@= 0. 

The hyperbola whose axes are a and 6 can be constructed 
by means of the equilateral hyperbola whose axis is a, just as 


CHAP. V. CONCERNING THE HYPERBOLA. 213 


the ellipse having the axes a and b was constructed by means 
of the circle of radius a; that is, the first hyperbola can be 
regarded as the orthogonal projection of the second. But this 
construction has no practical utility in the graphical construc- 
tion of the hyperbola, inasmuch as the trace of an equilateral 
hyperbola is not more simple than that of any other hyperbola. 


181. Let x and y be the co-ordinates of any point of the 
plane; consider the expression 
a bg 

Fm 

This polynomial is equal to zero for a point M belonging to 


the curve; if a point P starting from © travels along a line 
drawn parallel to the oe 


—1. 








’ . D 
axis AA! (Fig. 108), the term B P' P 
M 
remains constant, while the term 
. | 
ae ee ° A’ T \A x 
sr diminishes or increases, accord- ‘ 
ing as the point P approaches or 
: s’ R’ 
recedes from the y-axis. Whence * | 
Fig. 108. 


it follows that the polynomial has 
a negative value for every point situated between the two 
branches of the hyperbola, and positive for all other points of 
the plane. 


THE TANGENT. 
182. The equation of the tangent at the point M, whose 
co-ordinates are w and y, is 
ce, ier gh 
5) mee 
In order to construct this line, one can determine the point 7 
(Fig. 108), where it intersects the axis OX. _ if in equation (5) 
one make Y= 0, it becomes X = or =", this length OT 








can be found by a third proportional. 


214 PLANE GEOMETRY. BOOK III. 


183. The angular coefficient of the tangent has the value 
ee. b 


oy ee 
a 
Suppose that the point M describes the arc AD; at A the 
angular coefficient is infinity, and the tangent perpendicular 
to the transverse axis; as & increases, the angular coefficient 


diminishes constantly and approaches the limit oF the angular 


a 
coefficient of the asymptote OR; the angle M7X diminishes 
therefore from 5 t0 ROX; at the same time the value of OT 


diminishes from a to 0; whence it follows that the asymptote 
is the limiting position of the tangent, when the point of con- 
tact is indefinitely removed. 


184. To Draw a TancEent THROUGH AN EXTERIOR PoINT 
P. — If the co-ordinates of the point P be x, and y, the points of 
contact are determined by the equation of the chord of contacts 


(6) ct EDs Oy 


combined with equation (1) of the hyperbola. 
By eliminating y, one gets the equation of the second degree 


ao ey yy Ue va No 
gos) tt =o 


whose roots are the abscissas of the points of contact M and 
M' of the two tangents drawn from the point P. The condi- 
tion that the roots are real is 


that is, that the point P should be situated between the two 
branches of the curve. If the point P lie in the angles of the 

2 2 
asymptotes which embrace the curve, the coefficient 1-9 


being positive, the product of the roots is positive; conse- 


CHAP. V. CONCERNING THE HYPERBOLA. 215 


quently the two roots have the same sign, and the two points 
of contact on the same branch of the curve. On the contrary, 
if the point P be in one of the angles ROS, R'OS', there 
will be a point of contact on each of the branches. 


185. TANGENTS PARALLEL TO A GIVEN STRAIGHT LINE. — 
It is to be noticed that the equation of the hyperbola referred 
to its axes differs only from that of the ellipse in that 6° is 
replaced by — b?; if this change be made in equation (7) of 
§ 157 of the tangent to the ellipse, one gets the equation of 
the tangent to the hyperbola 


(7) y = mx + Varm — b*. 

In order that the problem be possible, it is necessary that 
2 

the value of m? be greater than ue that is, that in case the 
a 


given line passes through the origin, it lie within the angle 
ROS. It has already been shown (§ 183) that the numerical 


value of the angular coefficient of a tangent is greater than 0 
a 


186. One can draw to a hyperbola two perpendicular tan- 
gents so long as the angle ROR’ is less than a right angle, that 
is, when a is greater than b; when this condition is satisfied, 
the locus of the vertex of a right angle circumscribed about a 
hyperbola has the equation 


et y= a?—v?; 


that is, a circle concentric with the curve. 


DIAMETERS. 


187. When the hyperbola is referred to its axes, the diameter 
which bisects parallel chords whose angular coefficient is m 
has the equation 


2 @ 2my _ 9 
[es ee 
or nae 


216 PLANE GEOMETRY. BOOK III. 


If one designate by m' the angular coefficient of the diameter, 
there will exist, between the direction of the chords and that 
of the diameter, the relation 


2 


o np 2 


This relation shows that if one 











y ? take m! as the angular coefficient 

7; of the chords, one will find m 

Y Ziff" for the angular coefficient of the 

the Psi _ diameter; that is, in case the line 

: of-+ ~ DD" bisects the chords parallel to 

! . Ay x HE' (Fig. 109), reciprocally the 

D ir Ke -- line LE’ bisects the chords paral- 
Nfs te lel to DD'. Thus the two diame- 
Mie a ters DD', FE' are such that each 
ie ‘N’ bisects the chords parallel to the 
Fig. 109. other; they are two conjugate 


Tere 
The hyperbola has an infinity of systems ‘of conjugate 
diameters, since one can choose at will one of the diameters. 
Relation (8) shows that m and m!' have the same sign; if one 


9 ; b : 
suppose them positive, m varies from 0 to —, m! will vary from 
a 


0 to pe the diameter DD! revolves from O.A toward the asymp- 


tote OR, and the diameter KE' from OB toward the same 
asymptote. One sees thus, that of the two diameters, one 
always intersects the curve while the other never meets it. 
The axes form the only perpendicular system of conjugate 
diameters, and the angle included between the two conjugate 


° . T 
diameters varies from 9 to 0. 


It can be shown, as in the case of the ellipse, that the tangent 
FH at the point D of the hyperbola is parallel to the diameter 
EE", the conjugate of the diameter Dr which i is drawn to the 
point of contact (§ 160). 


CHAP. V. CONCERNING THE HYPERBOLA. yay 


188. Two conjugate hyperbolas and the system of their 
asymptotes possess the same diameter for the same series of 
chords; because the equations of the three loci 

2 2 2 
ey eee Vee 


ee -’ @ BF 
differ only by the constant term which does not enter in the 
equation of the diameter jf’, + mf’,=0. The three loci possess 
also the same systems of conjugate diameters. 


2 
If the hyperbola is equilateral, the relation mm! = becomes 
a 


mm'=1, which shows that the angles DOX, EOX are com- 
plementary, and, consequently, that the asymptotes are the 
bisectors of the angles of the conjugate diameters. 


189. THe HyprrRBoLA REFERRED TO Two CoNJUGATE 
Diameters. — When two conjugate diameters OD, OF (Fig. 
109) are chosen as co-ordinate axes, the equation of the hyper- 


bola becomes (§ 136) 
Al? t+ O"yY+ H=0. 


The coefficients A’ and C" have contrary signs, for example, 
C" has the sign of H, and A" the contrary sign; if one put 


at =, pra, 
the equation takes the form 
gy? 
(9) a" ra be = at , 


which is of the same form as that of the curve referred to its 
axes. 
Since one has, by the transformation of co-ordinates, 


for every point of the plane, it follows that the equation of the 
conjugate hyperbola, referred to the same diameters OD, OL, is 


12 12 


~~ P= 1. 


a’ b 12 


218 PLANE GEOMETRY. BOOK III. 


The diameter OE, which does not meet the first hyperbola, 
meets the second in the point EZ, and the length 6b! of this 
semi-conjugate diameter of the first hyperbola is equal to the 
length OE of the real semi-diameter of the second. 


190. Equation (3) of the asymptotes transforms into the 
equation 
dae apes 


b! 
5a Be 0, ee i agen 


One deduces, therefore, that the diagonals of the parallelogram 
FHGK, constructed on any two conjugate diameters, coincide 
with the asymptotes of the hyperbola. 

The sides FH, GK, of the parallelogram are tangents to the 
first hyperbola, and the sides FA, GH, to the conjugate, in 
such a way that the parallelogram is circumscribed to the 
curves of the two systems. 


THEOREM OF APOLLONIUS. 


191. It is sufficient to repeat the reasoning of § 163. 
By the formulas of transformation of co-ordinates, the two 
binomials 


aero a + ¥', 
are changed into 


12 12 
fF) gh + y+ 2 aly! cos 6. 


a” b 12 v 


The polynomial 


ee wee 
09 eG 


is transformed into 


gel? 


CHAP. V. .CONCERNING THE HYPERBOLA. 219 


The two polynomials (10) and (11) being perfect squares for 
the same values of A, the two equations 


(12) (1 — a’) (A +b) =0, 
and 
(13) 2 — (a” — b®)A — ab” sin’ 6 = 0, 


have the same roots; whence it follows that the two roots 
of equation (13) are equal respectively to a’ and — 0b’; one 
deduces the relations 


(14) a? — >? = a? — BY, 
(15) ab? sin? @ = ab’, or a’b' sin 0 = ab, 
and, therefore, the two following theorems: 


1° The difference of the squares of any two conjugate diameters 
is constant and equal to the difference of the squares of the axes. 


2° The area of the parallelogram constructed on two conjugate 
diameters is constant and equal to the area of the rectangle con- 
structed on the axes. 


It follows from the relation a? — b? = a?— 0’ that if a be 
different from b, one cannot have a!=b'; the hyperbola can- 
not have equal conjugate diameters. If, however, the hyper- 
bola be equilateral, one always has a'=6'; every system of 
conjugate diameters is equal; this agrees with the remark of 
§ 188, for then the two diameters make equal angles with 
the asymptote. 


192. Since the hyperbola and its two asymptotes have the 
same diameter for the same system of parallel chords, the mid- 
point I of the chord MM' is also the mid-point of the chord 
NN' (Fig. 109). Therefore the portions MN, M'N' of a secant 
comprised between the hyperbola and its asymptotes are equal. 

If the secant become tangent, one has DF = DH. The 
portions of a tangent comprised between the point of contact 
and the asymptotes are equal. 


220 PLANE GEOMETRY. BOOK III. 


193. Suppose that the hyperbola is referred to two con- 
jugate diameters DD', EE', of which HE’ is parallel to a 
given secant MM'; the curve will have the equation 


Db? : : 
and the asymptotes y” = ae In Fig. 109 the secant MM' 


intersects the same branch of the curve in two points, while the 
parallel diameter HE' does not meet the curve; and one has 


ee bh? 9 b'?— 
MI = ON —a"), NI =, OF", 
and, consequently, 
NI — MI’ = b*, or (NI— MI) (NI 4+ MI) =2°; 
but NI— MI= MN, NI+ MI= MN’; 
therefore MN -MN'=b"”. 


In case the secant intersects the two branches of the hyper- 
bola, the parallel diameter meets the curve, and one will arrive 
at an analogous result. Thus, the product of the segments of a 
secant, comprised between a point of the curve and the asymptotes, 
is equal to the square of the semi-diameter parallel to the secant. 


194. Being given the asymptotes RR', SS', and a point M 
of the hyperbola, one can obtain as many points of the curve 
as one wishes (Fig. 110). Draw, in fact, through the point M 
any straight line NMN'; this line 
intersects the asymptotes in WV and 
N'; if one take on this line a length 
N'M' equal to NM, one will have 
a second point M' of the hyperbola. 
= The direction and lengths of the axes 
R! Q may also be determined. The curve 
being comprised within the angles 

shia ROS, R'OS', the bisector OA of 
these two angles will be the transverse axis, and the perpen- 
dicular OB the imaginary axis. Draw QMQ' perpendicular 








CHAP. V. CONCERNING THE HYPERBOLA. 221 


to OA; the imaginary semi-axis 6 will be a mean proportional 
between MQ and MQ'. On laying off on OBa length OB equal 
to b, and drawing BC parallel to OA, BC will be the real 
semi-axis a. In order to construct a tangent at a point M’ of 
the curve, one will draw through this point M'P parallel to 
an asymptote, taking OG = 2 OP; the straight line M'G will 
be the tangent required. 


195. When one knows the positions and the magnitudes of 
two conjugate diameters, one can easily 
find the axes. Let, in fact, DD', HE' 
(Fig. 111) be the two diameters, of which 
the first is real. The diagonals of the 
parallelogram constructed on the two 
diameters are the asymptotes. Know- 
ing the asymptotes and a point D, one 
is led to the preceding construction. 





196. SupPLEMENTARY CHorps.— Two chords, MC, MC", are 
called supplementary chords if they, starting from the same 
point of the curve, be drawn to the 
extremities of the same diameter  \ 
CC' (Fig. 112). One can demon- 
strate, as has been done in § 168 
for the ellipse, that two supplemen- 
tary chords are parallel to a system 
of conjugate diameters, and that, 
reciprocally, if straight lines par- - 
allel to two conjugate diameters be 
drawn through the extremities of a diameter, these lines will 
intersect on the hyperbola and form a system of supplementary 
chords. 





Fig. 112. 


232. PLANE GEOMETRY. BOOK III. 


THE HYPERBOLA REFERRED TO ITS ASYMPTOTES. 


197. If, after having transferred the origin to the center, 
which makes the terms of the first degree disappear, one take 
for new axes of co-ordinates the two 
asymptotes OX, OY (Fig. 113), a 
line parallel to an asymptote will 
not meet the curve in more than one 


Z 
eo 
Cc 
M 
| <I < point; the equation should be re- 
Tr: 
~ 
X 














duced to the first degree in y and 
also in; that is, that the coefficients 
of 7’ and a are zero. The equation 
Fig. 113. will, therefore, have the form 














(16) 2 Biay + H=0, or sy =k. 


One deduces the value of k, on noticing that the co-ordinates 
of the vertex A are 


! fea 2 


which satisfy the equation of the curve; whence 


a? + ap 


k= 
a 


198. When the hyperbola is referred to its asymptotes, the 
tangent 77" at the point M, whose co-ordinates are’ @ and y, 
has the equation 


(17) yX +eY=2k. 


The abscissa of the point of intersection of the tangent with 
the axis OX is found by putting in this equation Y= 0, whence 


MOT en —9 OF 
y 
one has a second proof that the point of contact M bisects the 
portion 77" of the tangent comprised between the asymptotes 
(§ 192). 


CHAP. V. CONCERNING THE HYPERBOLA. 223 


THe AREA OF A HYPERBOLIC SEGMENT. 


199. Next is discussed the theorem which concerns the 
evaluation of areas. Consider the area bounded by the axis 
OX, a curve, a fixed ordinate AB, and a variable ordinate MP 


(Fig. 114), corresponding to the abscissa , 


«. This area, which is represented by S, ae 
is a function of the variable x, whose ‘WE 
derivative is to be determined. Give to a B 





an increment Aw = PP" sufficiently small 
so that the ordinate of M may vary in the one: oe 
same sense as that of M’. Draw through eer 

the points M and M', MC, M'D parallel | 

to the axis OX. The increment AS of the area is greater than 
the parallelogram MPP'C, and smaller than the parallelogram 
DPP'M'. The measure of the first parallelogram is yAz sin 0, 
6 being the angle between the axes, of the second (y+ Ay) Avsin 6. 
Therefore it follows 





yAx-sind< AS < (y+ Ay) Aw - sin 8, 
and, by dividing by Aa, 
ysing <5 < (y + Ay) sin 0. 


Let, now, Ax approach the limit zero. The ratio as les 
e 


between two quantities, the one y sin 6, the other having this 
quantity for its limit; therefore the ratio has also the same 
limit ysin@. Thus the derivative of the area considered as a 
function of the abscissa is ysin #6. Reciprocally, the area S is 
a function of y sin 6, considered as a function of x In case the 
axes of co-ordinates are rectangular, the derivative of the area 
is equal to y. 


200. Consider a hyperbola referred to its asymptotes, and 
determine the value of the area bounded by the asymptote OX, 
the hyperbola, the fixed ordinate AB corresponding to the 
abscissa a and the variable ordinate MP corresponding to the 


224 PLANE GEOMETRY. — BOOK III. 


abscissa # (Fig. 115). It follows from equation (16) that 


Y y= 2 and, consequently, 


8! =ysin@=ksing.— 





Since : is the derivative of log x; 


of 1 
therefore k sin 0- > is the deriv- 
ative of ksin@log «x; one has, 
consequently, 


Fig. 115. S=ksin 6 loga#+C. 


The constant C is determined by the condition that the area be 
zero for «=a, which gives C= — ksin @ loga. Hence, it fol- 
lows 


(18) S =k sin 6 (log # — log a) = k sin @- log (5) 
a 


The abscissa a being constant, if « be made to increase indefi- ~ 
nitely, the area S increases also without limit. The same 
occurs when a approaches the limit zero, « remaining fixed. 

In the particular case when the hyperbola is equilateral, one 
has sind =1; if in addition k be made equal to 1, and the 
‘area be reckoned from the ordinate which corresponds to the 
abscissa 1, that is, from the vertex of the curve, the preceding 
formula reduces to 

108 a: A 
It is on account of this property that Napierian logarithms 
have also been called hyperbolic logarithms. 

If one assume kK = 1, a= 1, formula (18) becomes 


S = sin 6 log 2. 


The angle 6 could be taken in such a way that S be the loga- 
rithm of « in any system whatever whose base is greater than e. 


CHAP. V. CONCERNING THE HYPERBOLA. 225 


EXERCISES. 


1. The base of a triangle is fixed; the difference of the 


angles at the base is z find the locus of the third vertex of 


the triangle. 

2. What is the locus of the centers of the circumferences 
which intercept given lengths on the sides of a given angle ? 

3. Being given two fixed straight lines and a movable 
straight line which intersects the first two in such a way that 
a triangle of constant magnitude is formed, it is required to find 
the locus of the centers of gravity of these triangles. 

4. Two secants drawn from any point of a hyperbola to 
two fixed points taken on the curve intercept on one or the 
other asymptote constant lengths. 

5. Every chord of a hyperbola bisects the portion of one 
or the other asymptote comprised between the tangents at 
its extremities. 

6. If, ona chord of a hyperbola considered as a diagonal, one 
constructs a parallelogram whose sides are respectively parallel 
to the asymptotes, the other diagonal passes through the center. 

7. Being given a fixed point and a fixed straight line; an 
angle of constant magnitude rotates about its vertex placed at 
the fixed point; find the locus of the center of the circle circum- 
scribed about the triangle formed by the sides of the angle and 
the fixed straight line. 

8. A triangle ABC is inscribed in a hyperbola; two of its 
sides have fixed directions; find the locus of the mid-point of 
the third side. 

9. On one of the diagonals of a rectangle used as a chord a 
circle is described; find the locus of the extremities of the 
diameters parallel to the second diagonal. 

10. Being given an angle and a fixed point, one draws through 
this point an arbitrary secant, and through the points in which 
this secant intersects the two sides of the angle, one draws 
straight lines respectively parallel to these sides; find the 
locus of the point of intersection of these parallels. 


11. Find the locus of a point such that on drawing through 
FP 


226 PLANE GEOMETRY. BOOK III. — 


this point lines parallel to the asymptotes of a hyperbola, the 
area of the triangle formed by these parallels and the hyper- 
bola is equal to a given constant. 

12. Find the locus of a point such that one of the bisectors 
of the angles formed by the straight lines which join this point 
to two fixed points A and B has a given direction. 

13. Every equilateral hyperbola circumscribed to a triangle 
passes through the point of intersection of the altitudes. 

14. Being given an ellipse, one draws any two conjugate 
diameters ; find the locus of the point of intersection of one of 
them with a straight line drawn through a fixed point perpen- 
dicular to the other, or, more generally, with a straight line 
making a given angle with the second diameter. 

15. Being given two straight lines A'A and B'B and the 
point 0; about the point O as center, with an arbitrary radius, 
a circle is described ; at the points of intersection of the circle’ 
with the straight lines perpendiculars are erected to these lines ; 
find the locus of the points of intersection of these perpen- 
diculars. 


CHAP. VI. CONCERNING THE PARABOLA. 22 


CHAPTER VI 


CONCERNING THE PARABOLA. 


201. The second type to which the equation of the second 
degree may be reduced is Cy’? + 2 D'z = 0, or 


(1) a? = 2 px. 


The case when p is negative can be treated under the case 
when p is positive by reversing the direction in which one 
measures the positive abscissas; assume therefore that p is 
positive. It follows immediately from the form of equation 
(1) that the curve is symmetrical with respect to the axis of a, 
and that it passes through the origin. Equation (1), solved 
with respect to y, gives 


y= tV2 ve. 


In order that the ordinate be real, it is necessary that the 
abscissa be positive; if 2 increase from 0 to + #, the absolute 
value of y increases also from 0 to 
©; thus it follows that the parabola 
consists of two infinite arcs AD 
and AD! (Fig. 116). 

The straight line AX is the 
axis of the parabola, the point A 
is the vertex, the length p, which 
determines the magnitude of the 
curve, is called the parameter of 
the parabola. 


202. Construction of the curve 
by points. The ordinate MP of 
the point M is a mean proportional between the constant 
length 2p and the abscissa AP. Construct on AX, in the 
direction of negative abscissas, a length AQ equal to 2p; 











Fig. 116 


228 PLANE GEOMETRY. BOOK III. 


then describe diverse circumferences whose centers lie on QX, 
and pass through the point Q; these circumferences intersect 
the axis AX in the points P, P', ---, and the line AY in the 
points N, N', ---. Through the points P, P’, .--, draw lines 
parallel to AY; through the points NV, N’, ---, lines parallel to 
AX; their points of intersection M, M', ---, belong to the 
parabola. 


203. From the relations 
MP’ =2p-AP, M'P?=2yp.AP, 
Arm 

one deduces Sl a 
mM Pp" AP'! 

The squares of the ordinates perpendicular to the axis of a 
parabola are proportional to the segments of the axis comprised 
between the vertex and the ordinates. 


204. Through the point M of the curve, draw a parallel to 
the axis, and imagine that a movable point travels along this 
parallel. Replace in the function 7? — 2 pa, x and y by the co- 
ordinates of the movable point; if the point J be situated on 
the side of the positive abscissas with respect to the parabola, the 
function will be negative, if the point M be on the other side, 
the function will be positive. For brevity the first region is 
said to be interior and the second eaterior to the curve. 


205. It. has already been shown that the infinite -branches 
of the hyperbola have asymptotes; the same is not true of the 
parabola. For, since y increases indefinitely with a, there 
- cannot be an asymptote parallel to the axis of the parabola. 

In the second place, let y=ax+b be the equation of any 
straight line oblique to the axis, the difference of the ordinates 
of the points of the line and of the curve which correspond to 
the same abscissa is equal to 

av +b —V2 pa, 
and can be put under the form 


. oa +b—aP?), 
a Ay 


CHAP. VI. CONCERNING THE PARABOLA. 229 


When @ increases indefinitely, the first factor increases indefi- 
nitely, and the second approaches the value a different from 
zero, the product increases indefinitely. Therefore an asymp- 
tote oblique to the axis cannot exist. 


TANGENT. 


206. The tangent at the point M, whose co-ordinates are x 
and y, has the equation 


(2) yY=p(X +2). 


Let 7 be the point where the tangent intersects the axis of 
the parabola (Fig. 117); if in equa- 
tion (2) one make Y=0, then will 
A =—x; therefore AT= AP. This 
property furnishes a means for con- 
structing the tangent to the parab- 
ola at a given point M; to construct 
the tangent draw MP perpendicular 
to the axis, take A7’= AP, and con- 
nect the points M and P with a 
straight line. 











Fig. 117. 


207. To draw a tangent through an exterior point M,. Let 
a, and y, be the co-ordinates of the point M,; the points of 
contact will be determined by the chord of contact 


(3) ny = p(@ +m), 


combined with that of the curve (1); whence it follows 


2 


Y="HtvVy" — 2px, cal re 


these values are real so long as the point M, is exterior to the 
curve. | 

In order to construct the line MM’, one seeks the points 
where it intersects the co-ordinate axes; if, in equation (3), one 
put y=0, one gets x = —a,, whence AJ is equal to AP,; if one 


230 PLANE GEOMETRY. BOOK III. 


put «= 0, one finds y=”; the point K may be found by a 


fourth proportional. ; 


208. To draw a tangent parallel to a given line. If m rep- 


resent the angular coefficient of the given line, the equation 
77 m, and that of the curve, determine the co-ordinates of 
the point of contact, y = a 1 ome sos It follows then that the 
equation of the tangent will be 


pap Gia ae 
(4) : Y=mX +5 


209. Norma. —The normal MN ata point M of the parab- 
ola, whose co-ordinates are « and y, has the equation 


(5) Y—y=— 7 (X-2), 


On putting Y=0, one obtains the abscissa of the point NV 
where it intersects the axis; one finds 


Thus, in the parabola the sub-normal PN is constant and equal 


to the parameter p. 


DIAMETERS. 


210. By applying the general equation of the diameters of 
a curve of the second degree to the parabola, whose equation 
is y? — 2px = 0, one obtains the equation 

) 
(6) my —p=0, or y= e. 
This property has already been demonstrated in § 134; it is, 


that every diameter of the parabola is parallel to the axis. 
Since the angular coefficient m of the chords can be so chosen 


that z can take any value that one chooses, it follows that, con- 


versely, every straight line which is parallel to the ais is a 
diameter. 


CHAP. VI. CONCERNING THE PARABOLA. 931 


Let A’ be the point of intersection of the diameter with 
the curve (Fig. 118); since the ordinate of the point A’ is 
Pp “ i 


Y oF, 








equal to = and the angular coefficient of Z 

the tangent at this point has the value Pie Sate 
A j a“ D 

. that is m, it follows that the tangent at a a 

the extremity of a diameter is parallel to "+, 


the chords which this diameter bisects. 





211. Parabola referred to one of its 
— diameters and to the tangent at its ex- 
tremity. It has been proven (§ 136) Fig. 118. 

that, in case a diameter A/X' and the tangent A'Y’ at its ex- 
tremity be taken as the axes of co-ordinates, the equation of 
the parabola will have the form 


(7) y? = 2 pia. 


If a and b be the co-ordinates of the point A’ with respect to 
the primitive axes, and AP!’ be drawn parallel to A'T, one 
knows that one has A'P! = AT'= AP (§ 206); the co-ordinates 
A'P', — A'T of the vertex A with respect to the new axes are 
therefore a and —V4a?+ 6"; since they satisfy equation (7), 
it follows that 











2 2 2 
Qo! = 2% atl Ade a a ol 
a a 
re) Ary 
One has also p!= Aon ep == EN, 


If the angle Y'A'X' formed by the new axes be represented by 
6, it follows from the right triangles NA'T, NA'P, that 


AN any = PN 


TN 
sin’ sin 0 





PN p 
whence '_. PN — oat at 
Pe en nip 


pad PLANE GEOMETRY. BOOK III. 


212. Since the parabola, referred to a diameter A'X and 
to the tangent A'Y (Fig. 119), has the equation y? = 2 p'ex, it 
is evident that the equation y ¥ = p'(X + «) represents either 
the tangent at the point M, if # and y be 
the co-ordinates of this point, or the chord 
of contact of the tangents drawn from an 
exterior point whose co-ordinates are x and y. 

The tangents at the two extremities M 
and M' of a chord intersect the diameter 
in the same point 7, such that A'T’ = A'L. 
It is also true that the chord of contact 
MM', with respect to an exterior point 7, 
is bisected by the diameter 7'X which passes 
through this point, and for a greater reason 
ia A 

This furnishes the means for constructing a parabola by 
points, in case one knows two tangents 7'M, TM’, and the 
points of contact M and M’. Draw the chord MM’, and join 
the mid-point J with the point 7, the mid-point A’ of the 
straight line 77 is a point of the curve, and the tangent at 
this point is parallel to MM'. By means of the tangent A'7", 
which touches the curve at A', and of each of the given tan- 
gents, one can determine two new tangents by their points of 
contact, and so on. This method for constructing two parallel 
lines by means of an are of a parabola is frequently used, 
when the are of a circle cannot be employed; that i is, When the 
distances 7M and 7M’ are not equal. 








Fig. 119. 


THE AREA OF A PARABOLIC SEGMENT. 


213. It is proposed to evaluate the area S of the triangle 
A'IM formed by the straight lines A'/, 1M and the are A'M 
of the parabola (Fig. 119). If this area be regarded as a 
function of the abscissa of the point M, the derivative S' is 
given by the formula 


SS ys 0 = V2 as sin 6 =V2p'-sin +a. 
One deduces S=ZV2p'-sind-2? + C. 


CHAP. VI. CONCERNING THE PARABOLA. 233 


The constant C is zero since the area becomes zero for «= 0. 
It follows therefore that 


S=22-V2p'v-sin 6 = 2 aysin 6. 


The area S is equal to two-thirds of the parallelogram A'IMN, 
and, consequently, the area of the composite-line triangle 
A'NM is one-third of the same parallelogram. 


EXERCISES. 


1. Find the locus of the vertex of an angle circumscribed 
about a parabola, such that the triangle formed by the sides 
of the angle and the arc of the parabola has a constant area. 

2. Find the locus of points from which two perpendicular 
normals can be drawn to a parabola. 

3. A secant revolves about a fixed point taken on the axis 
of a parabola; normals are drawn to the parabola at the 
points in which the secant intersects it; find the locus of 
the point in which these normals intersect. 

4. A parabola moves parallel to itself, so that its vertex 
traces the parabola in its initial position; tangents are drawn 
from the vertex of the fixed to the movable parabola; find the 
locus of the points of contact. 

5. Find the locus of a point from which*the sum of the 
squares of the normals drawn to a parabola is constant. 

6. Given a curve of the second degree tangent to the sides 
of a given angle, one draws an arbitrary tangent to this curve; 
find the locus of the point of intersection of the medians or 
the altitudes of the triangle formed by the movable tangent 
and the sides of the angle; find also the locus of the center 
of the circle circumscribed about this triangle. 

7. Given an ellipse, one draws through a fixed point any 
two straight lines at right angles to each other, and at the 
points in which these lines intersect the ellipse, tangents are 
drawn to this ellipse; find the locus of the points of inter- 
‘section of these tangents. 

8. Same problem, when one replaces the perpendicular 
lines by lines parallel to the conjugate axes of any other 
given ellipse. 


934 PLANE GEOMETRY. BOOK Itt. 


9. An angle of constant magnitude revolves about its 
vertex situated on a given curve of the second degree; at the 
points in which the sides of the angle meet the curve again, 
tangents are drawn to the curve. Find the locus of the point 
of intersection of these tangents. 

10. Find the locus of the center of an equilateral triangle 
formed by three tangents or by three normals to a parabola. 

11. The area of a triangle whose vertices are the points of 
contact of three tangents to a parabola is twice the area of the 
triangle formed by these tangents, and is represented by the 
expression 


a8 att i eer Ht 
£70 yy" — y")Gyl"—y), 


where 7, y", y'"’ represent the perpendiculars dropped from 
the vertices of the triangle to the axis. 

12. An arbitrary tangent is drawn to a hyperbola, and the 
points in which the tangent meet the asymptotes are respec- 
tively joined to two fixed points; find the locus of the point 
of intersections of the two straight lines. 

13. Draw to a parabola a normal so that the area comprised 
between this normal and the curve has a minimum value. 


CHAP. VII. FOCI AND DIRECTRICES. _ 285 


CHAPTER VII 


FOCI AND DIRECTRICES. 


215. The discussion is begun by proposing the following 
question: Given a point F and a straight line DE (Fig. 120), 
find the locus of a point whose distances 
from a given point and a given straight line 
are in a constant ratio. 

Draw in the plane any system of rec- 
tangular axes; call « and £ the co-ordinates 
of the point /, and let mxa+ny+h=0 be 
the equation of the line DE; the distances 
of any point M of the plane from the point 
F and from the line DE are given by the 














formulas Fig. 120. 
: + (ma + ny + h) 

MF = —_ 2 pate 2 MP as . 

V (a — a)? + (y — B)’, ae } 


if the constant ratio be designated by k, the locus will 


have the equation 











' k(ma + ny + h) 
—— + wee 2 Leong SAE ETN ? 
k?(ma + ny + h)? 
2 yj iat 
or (w me (t) a (y ae B ) ii m2 ee n? x 


This locus is a-curve of the second degree. The quantity 
AC — B’, which serves to distinguish the species of the curve, 
being equal to 1 — k?, the curve is an ellipse, a parabola, or a 
hyperbola, according as the ratio k is less, equal to, or greater 
than unity. 

Conversely, given a curve of the second degree one proposes 
to seek if there exist in the plane of the curve a fixed point # 


236 PLANE GEOMETRY. BOOK III. 


and a fixed straight line DH, such that the ratio of the dis- 
tances of each of the points of the curve from the point F and 
the line DE is constant. If one find a point and a straight 
line enjoying this property, the point will be called the focus 
of the curve, and the straight line the directriz. 

The axes of co-ordinates being arbitrary and inclosing an 
angle 6, suppose that one has found a point # whose co-ordi- 
nates are « and £#, and a straight line DE whose equation is 


mx + ny +h=O0, such that the ratio ah is equal to a constant 


quantity k; since the distance MP of a point M of the curve 
whose co-ordinates are x and y from the directrix DE is repre- 
sented by the expression 


+ (mx + ny + h)sin @ 
Vm? + n? — 2 mn cos ra 








one will have 
k(mex + ny + h) sin 6 


MF=+ 
Vm? + n? — 2 mn cos 6 








Thus the distance of any point W of the curve from the focus 
F is expressed as an integral function of the co-ordinates « and 
y of the point M and is of the first degree. 

Conversely, if a point F’ enjoy the property that its distance 
from any point M of the curve is expressed by an integral 
function of the first degree in the co-ordinates x and y of the 
point M, this point F is the focus; that is, that there exists a 
straight line DE such that the ratio of the distances of each 
of the points of the curve from the point F and the line DE is 
constant. In fact, assume that one has 


FM=+(m«e+ny+h), 


where mx + ny +h represents an integral function of the first 
degree in the co-ordinates x and y of the point M. Consider 
the straight line DH which has the equation 


me+ny+th=0; 


CHAP. VII. FOCI AND DIRECTRICES. 237 


the distance of the point M from this line is given by the 
formula 





MP — + (ma + ny + h) sin 6 
Vm? + n? — 2mncos 6 





one has therefore 
MF _Vm?+n?—2mncosd 
MP sin 6 








Thus the ratio of the distances of each of the points of the 
curve from the fixed point F and the fixed line DE is constant; 
the point F’ is therefore a focus and the line DE the cor- 
responding directrix of the curve. Representing the value of 
this constant ratio by k, one has | 





k sin 6 =Vm? + n?— 2 mncos 0. 


216. Therefore the following definition can be substituted 
for the first. The focus is a point such that its distance from 
any point of the curve can be expressed by an integral function 
of the first degree in the variable co-ordinates of a point of the 
curve. It is clear, moreover, that this algebraic definition is 
independent of the position of the co-ordinate axes in the plane, 
because an integral function of the first degree preserves its 
character in case the axes are changed. The equation of the 
directrix is found by equating this function to zero. 

If the y-axis be taken parallel to the directrix, the vaxis 
being arbitrary, the equation of the directrix will take the form 
mx +h=0, the coefficient n will be zero and the distance 
of the focus from any point M of the same will be expressed 
_by an integral function + (ma +h) of the first degree in the 
abscissa # of the point J. 

From what precedes it follows that the investigation of the 
focus and the directrix of curves of the second degree is 
reduced to the determination of a point F, such that its 
distance from any point M of the curve is expressed by an 
integral function of the first degree in the co-ordinates x and y 
of the point M. Suppose that the axes are rectangular, and let 


(1) Aa? + Bey + Cy’? +2 De+2 LEy+ F=0 


238 PLANE GEOMETRY. BOOK IIt. 


be the equation of the given curve of the second degree. Call 
« and £ the co-ordinates of the focus sought; then will the 
co-ordinates of every point of the curve satisfy the equation 





V (x — a)? + (y — B)? = + (ma + ne + hi), 
or (2) (w — a)’ + (y — B)? — (me + ny +h) =0. 


Equations (1) and (2), representing at the same time the same 
curve, are identical, hence the coefficient of corresponding 
terms must be proportional; one will have, therefore, to de- 
termine the five unknown quantities «, B, m, n, h, the five 
equations 

















(3) fe a mh) 
A B C D : 
_—(B+nh) @+P?-W 
= a = “a : 


In order to simplify the calculation, one considers separately 
the three curves of the second degree, referred to systems of 
rectangular axes which have served to simplify their equa- 
tions. Later will be given another method for finding the 
foci, especially useful for finding the geometrical loci of the 
foci. 


Foct AND DIRECTRICES OF THE ELLIPSE. 


217. Let 
2 
(4) aia ee 


be the equation of the given ellipse referred to its axes. This 
equation does not have any term in ay; it is necessary there- 
fore that the coefficient —2 mn of the corresponding term in 
equation (2) be zero, whence it follows that one has either 
n=0,orm=0. Suppose thatn=0; since the terms of the 
first degree are also zero, one will have a+ mh=0, B=0O, 
and equations (3) reduce to 


a (1 — m?) = 0? = h? — &’. 


CHAP. VII. FOCI AND DIRECTRICES. 239 


2 


2 
a* — : 
Whence one deduces m? = ; since m can always be sup- 





posed positive, without changing the signs of the coefficients 





: : Va? — 6? ; 
m, n, h in equation (2), one takes Noa ame If in the 


equation a?(1 — m?) =h?— a, h be replaced by its value de- 
duced from the equation «+ mh=0, one gets e=a’?— Dd’, 
whence e=+Va?— 02, h= Fa. 

Thus are obtained the two foci # and F" (Fig. 121), situated 
on the major axis at equal distances from the center. In order 
to determine them, one describes with a radius equal to a about 
the extremity B of the 
minor axis as center, a # 
circle; the points F# and 
F' where this circle inter- 
sects the major axis, are 
tue foci. If, for brevity, oT, . 2 
one put a?— 0?=c’, one 


“ye E 

















é 
HAS «== + ¢, aa 


‘Fig. 121. 


h=-7a; the upper signs 

correspond to the focus F, the lower signs to the focus F". 
One knows that the equation of the directrix is found by 
equating to zero the polynomial mv+ny+h; this equation 


2 . 
reduces to <2 Qe 0, OF 27 — Thus are obtained the 
2 
two directrices; the directrix whose equation is n= corre- 
sponds to the focus F, and the directrix whose equation is 
2 


a : . 
aa corresponds to the focus F". These directrices are 


perpendicular to the major axis and at equal distances from 
the center; the determination of the point D depends upon a 
third proportional; one constructs it in the following manner: 
describe on the major axis as diameter a circle, draw through 
the focus F' a perpendicular to this axis and, at the point V 
where this perpendicular meets the circle, draw a tangent to 
the circle; the point in which this tangent intersects the major 
axis is the point D. 


240 PLANE GEOMETRY. BOOK IT. 


It has also been seen that the constant ratio of the distances 
of each of the points of the curve from the focus and from the 
corresponding directrix is equal to Vm?+n?, in rectangular 


c eae 
co-ordinates; one has therefore k=m=-- The ratio — is 
a a 


called e, the eccentricity of the ellipse. 


218. Suppose now m= 0; the coefficients of the first degree 
should be zero; one will have e=0, 8+ nh =0, and equations 
(3) reduce to 

a’? = 6° (1 — n*) = hh? — B’. 


Whence may be deduced 


nate B=+vV0?—a, h=F bd. 

In order to obtain these new solutions, it suffices to permute 
in the first solutions the letters a@ and 6, m and n, « and £. 
Since a has been supposed greater that b, these two solutions 
are imaginary. Thus one can assign to the constants four 
systems of values which render equations (2) and (4) iden- 
tical; but two only of these systems of values which give the 
foci and the directrices are real. 


219. THEorEM I.— The swum of the distances of each of the 
points of an ellipse from the foci is constant. 
The distance of a focus from any point M of the*curve is 


expressed by +(max + ny +h), that is +(a— = ; the sign is 


so chosen that the quantity will be positive. The abscissas a 
and « of the focus, and of a point of the ellipse being less 
than a in absolute value, and, consequently, the quantity 
within the parentheses is positive for every point of the 
ellipse; it will be necessary, therefore, to give the parentheses 
the sign +, and one will have 


MF=a—ex, MF'=a+er; 


whence it follows 
MF + MF'=2a. 


CHAP. VII. FOCI AND DIRECTRICES. 241 


220. CorottAry I.— The sum of the distances of a point 
within the ellipse from the foci is less than the major axis; the 
sum of the distances of a point without is greater than the major 
Maxis. 

Consider first the point N (Fig. 122), situated within the 
ellipse, join this point to the two foci, and prolong the straight 
line FN till it intersects the ellipse in M@. Since the point 
M belongs to the ellipse, the sum of the two radii vectors 
MF + MF’ is equal to the major axes 4.4’; but the straight 
line NF is shorter than the broken line NM+ MF; by adding 
to each of these expressions the same length F'N, it follows 
that the path "N+ NF is shorter than F'M + MF, that is, 
less than AA’. Consider next a point P situated without the 
ellipse; the line PF" intersects the ellipse at a point I. The 
broken line MP + PF is greater than the straight line MF; on 
adding to both the same length F"M, one sees that the path 
F'P+PF is greater than "M+ MP, 
that is, greater than AA’. It is clear M 
that the converse propositions are true. 
If the sum of the distances of a point , , 
of the plane from the two foci be less ois : 
than the major axis, this point will lie ee. 
within the ellipse. If the sum be greater — 

; Cae ig. 122, 

than the major axis, the point will lie 

without. Whence it follows that one can consider the ellipse 
as the locus of the points of which the sum of the distances 
from the two foci is equal to 2a. Thus is the ellipse con- 
structed in elementary geometry, and it is on this property 
that the construction of the ellipse by points depends, or on a 
continuous motion, of which mention has been made at the 
beginning (§ 11). 


P - 





221. Cororiary II. —The ellipse in the locus of points equally 
- distant from the focus F and the circle described about the other 
focus F" as center with a radius equal to the major ais. If the 
foci be joined to any point M of the ellipse with straight lines 
(Fig. 123), and if the radius vector FM be prolonged till MH 


is equal to MF, one obtains a constant length F’H equal to the 
Q 


242 PLANE GEOMETRY. BOOK III. 


major axis; the locus of the point H is therefore the circum- 
ference described about the focus F’ as center with the major 
axis as radius. The portion MH of the radius being the shortest 
path from the point M to the circumference, the point M of the 
ellipse is equally distant from the focus ¥ and the circumfer- 
ence. The name director circle has been given to this circle. 





Fig. 123, 


222. TurorEM II.— A tangent to the ellipse makes equal 
angles with the radii vectores, which are drawn from the point 
of contact to the foci. 

Take two points M and M' (Fig. 124) on the ellipse; about 
the focus F' as center, with FM as radius, construct the are of 
a circle which intersects the radius vector FM" at O; the 

length M'C represents the difference 

A of the two radii vectores 7M and FM", 

or the increment which the radius vec- 
tor F'M receives when the point M has 
been moved to the neighboring point 
M'. Similarly, if one describe about 
the focus F' as center with the radius 
F'M an arc of a circle which intersects 
in D the radius vector #'M' produced, 
the length M'D will represent the difference of the two radii 
vectores F’M and F'M', or the negative increment which the 





Fig. 124. 


CHAP. VII. FOCI AND DIRECTRICES. 243 


radius vector F'M receives when the point M@ has been moved 
to the point M@’. Thus, when the point M moves to the point 
M', the radius vector F/M is increased by the increment M'C, 
while the other radius vector F’M is diminished by the inecre- 
ment M'D. Since the sum of the two radii vectores M+ F'M 
remains constant, the quantity by which the one is increased is 
equal to that by which the other is diminished, and, conse- 
quently, the two lengths M'C and M'D are equal. 

Draw through the two points M and M' the secant MS; 
draw in the two circles previously constructed the two chords 
MC and MD. Lay off on the secant MS the arbitrary but 
invariable length MG, and through the point G draw GH 
parallel to MC, G& parallel to MD; from the preceding 
construction it follows that one has the equal ratios 


M'C M'M_M'D. 
MH MG M'R’ 


since the two lengths M'C and M'D are equal, it follows that 
the two lengths M'H and M'K are also equal. Suppose now 
that the point WM’ approaches continu- 
ally the point M; the secant MS will 
approach a limiting position MT (Fig. 
125), which is a tangent to the ellipse. 
The points C and D will at the same 
time approach the point ©, the chords 
MC and MD, prolonged, approach the 
tangents to the circles described about 
the points # and F" as centers with 
FM and F'M as radii, and, consequently, become perpendicular 
to the radii FM and F'M; their parallels GH and GK take 
also directions perpendicular to the same radii, and, conse- 
quently, the angles H and & become right. The limits of the 
two triangles M'GH, M'GK (Fig. 124) are two right-angled 
triangles MGH, MGK (Fig. 125); these two triangles, having 
the common hypotenuse MG and the sides MH and MK, are 
equal, since they are the limits of equal lengths; whence it 
- follows that the two angles GMH, GMK are equal. There- 
fore the tangent MT to the ellipse bisects the angle FM/t 











Fig. 125. 


244 . PLANE GEOMETRY. BOOK III. 


formed by the radius vector MF and the prolongation of the 
other F'M. 

The vertical angles F'MT' and GMK being equal, one sees 
that the tangent 7'7" makes, with the two radii vectores drawn 
from the point of contact, the equal angles FMT, F'MT". 


223. CoroLtuAry J. — At the point M 

a4 (Fig. 126) draw to the tangent 77" a 

| “ perpendicular MN; it will be a normal 

to the ellipse. The two angles FMN, 

'MN are equal, since they are the com- 

plements of the equal angles FM f i 

F'MT"'; thus, the normal to the ellipse at 

wee the point M bisects the angle FMF" formed 
by the radii vectores which are drawn from this point to the two 


foci. 


a 


224. CorotiARy II.— Suppose that a light be placed at the 
He F (Fig. 127) of an ellipse; the rays of light, emanating 
from the point F, are reflected on the 

~ 7 ellipse, making the angle of reflection 
: equal to the angle of incidence. Let 
als : ei F'M be one of these rays; draw to the 
ellipse at this point the tangent 77"; 

ce, the reflected ray, which makes with M7" 
Fig. 127. an angle equal to WMT, will he reflected 

along MF’. Thus the reflected rays will 

all be concurrent at the second focus #", where they form a 
very brilliant image of the flame placed at the first focus F. 
It is on this account that the points F’ and Ff” are called 


foci. 


V gf 









225. Corotiary III.— Conversely, the ellipse is the only 
curve which enjoys the property that the radii vectores which 
are drawn from the point of contact to the two fixed points 
and F' and make equal angles with the tangent. Seek, in 
fact, the equation of the curve in bi-polar co-ordinates (§ 4), - 
and represent by wu and v the radii vectores MF, MF' (Fig. 


CHAP. VII. FOCI AND DIRECTRICES. 245 


124). When the point M of the curve moves to the point MM", 
the two radii vectores wu and v receive the increments, 


Au=+M'C, Av=— M'D, 


ay MD ME 
Au MC MH 


and one has 


When the point M’ approaches indefinitely the point M, the 
straight line MM' becomes a tangent and the two angles at 
H and K, as has been stated, become right. One supposes, 
moreover, the two angles GMH, GMK (Fig. 125) equal to 
each other; the two right triangles GMH, GM&K are therefore 


equal; one has MH = MK, and the ratio a approaches a 
U 


limit equal to —1. If one consider v as a function of wu, one 
sees that the derivative of this function is equal to —1; on 
returning to the primitive function, one has v= —w-+ C, and, 
consequently, w+v=C. Therefore the curve is an ellipse. 


226. CorottAry IV.— The locus of the projections of the 
foci on the tangents to the ellipse is the circle described on the 
major axis as a diameter. Prolong the radius vector F'M 
till MH is equal to MF; the tangent 
bisecting the angle PMH is perpendic- 
ular to the straight line PH at its mid- 
point (Fig. 128); join this point to the 
center O of the ellipse. The straight 
line OJ, which bisects the two sides 
FF', FH of the triangle F'FH, is paral- 
lel to the third side #"H, and equal to 
its half; the length #"H being equal ae arial 
to the major axis AA’, the distance OJ is constant and equal 
to OA. Therefore the locus of the point J is the circum- 
ference of the circle described about the point O as center, 
with OA as radius. 





246 PLANE GEOMETRY. BOOK III. 


227. Prosptem I.— To draw a tangent to an ellipse at a 
given point M on the ellipse. 

This problem has already been solved, by considering the 
ellipse as the projection of a circle. The same questions will 
be treated by another method which is 
gee ““! applicable to the hyperbola and the 

7 parabola. 

Prolong the radius vector F'M (Fig. 
129) till MH is equal to the other radius 
vector MF, and draw through the. point 
M a straight line J'7" perpendicular to 
FH; one will have the tangent required. 

Because in the isosceles triangle FMH, the line MT, drawn 
from the vertex perpendicular to the base FH, bisects the 
vertical angle. This line, being the bisector of the angle 
F'MH formed by one of the radii vectores and the prolongation 
of the other, coincides with the tangent to the ellipse. 





Fig. 129. 


228. Remarxk.— One should notice that all of the points 
of the tangent, excepting the point of contact M, lie without 
the ellipse. Let P be any point of the tangent; join this 
point to the foci and to the point H. The tangent being per- 
pendicular to FH at its mid-point, the distance PF is equal to 
PH, and, consequently, the broken line #’P+ PF is equal to 
the broken line /"P + PH; but the latter is greater than the 
line FH, which is equal to the major axis of the ellipse, since 
the radius vector MF" was prolonged till MH is equal to MF. 
Since the sum of the distances of the point P from the foci 
is greater than the major axis, this point is situated without 
the ellipse. 

The broken line #'M+ MF is the shortest path going from 
the point #" to a point.on the tangent and then to the point F. 

A broken line is said to be convex, in case it is situated on 
the same side with respect to each of its sides indefinitely pro- 
longed. Similarly, a curve is said to be convex in case it lies 
entirely on the same side of every tangent to it indefinitely 
produced. Accordingly, it follows that the ellipse is a closed 
convex curve. 


CHAP. VII. FOCI AND DIRECTRICES. 247 


229. ProsiEM II.— To draw to an ellipse a tangent from an 
external point P. 

Assume that the problem is solved, and let PM (Fig. 130) 
be a tangent passing through the point P. If the radius 
vector F'M be prolonged till MH is equal to MF, it follows 
that the tangent PM is perpen- 
dicular to the straight line FH at 
its mid-point; it remains there- 
fore to determine the point #. 
Since the line F’'H is equal to the 
major axis AA’, the point H is on 
the circumference described about 
the focus F’ as a center with AA! 
as a radius. On the other hand, 
the distance PH being equal to PF, the point H is on the cir- 
cumference described about the point P as a center with PF 
as a radius; the point H is therefore the intersection of these 
two circumferences. The following construction may be in- 
ferred from the preceding: Describe about the focus F" as 
center, with a radius equal to the major axis, a circle. De- 
scribe about the point P as center, with a radius equal to the 
distance PF of this point from the other focus, a second circle, 
which intersects the first in H. Join F and H by a straight 
line and draw from the point P a perpendicular to FH; the 
perpendicular will be the tangent required. The point of con- 
tact M will be determined by the intersection of the tangent 
with the line /'H. The two circles intersect in a second point 
H'; on drawing from the same point Pa perpendicular to FH", 
a second tangent PM' will be determined, whose intersection 
with the straight line F’H' will be the point of contact M’. 

These constructions can be accomplished without drawing 
the ellipse. It is sufficient that the foci and the major axis 
be known. 





Fig. 130. 


230. Prosiem III. — To draw to an ellipse a tangent which 
is parallel to a given straight line KL. 

Assume the problem to be solved, and let ST be a tangent 
parallel to KL (Fig. 131). If F"M be prolonged till MH is 


248 PLANE GEOMETRY. BOOK III. 


equal to MF, one knows that the tangent is perpendicular to 
FH at its mid-point. Whence the following construction can 
be inferred: Describe a circle about 
the focus F" as center, with a radius 
_ equal tothe major axis; draw through 

the other focus Fa straight line FH 
perpendicular to the given line KL; 
this line will intersect the circum- 
ference in a point H; draw ST 
perpendicular to FH at its mid- 
point; ST will be the tangent re- 
quired. The point of contact will 
be determined by the intersection 
oe of the tangent with the straight 
line F’H. The straight line FH, 
prolonged, intersects the circumfer- 
ence in a second point H'; on erecting a perpendicular to FH!’ 
at its mid-point, a second tangent S'7" will be found, whose 
point of contact M' will be determined by the intersection of 
F'H' with T'S". 





Fig. 131. 


231. Prositem IV.— An ellipse is defined by its foci and its 
major axis. Determine the points of its intersections with a given 
straight line MM". 

Let M be one of the points where the given straight line 
intersects the ellipse (Fig. 132); connect this point by, straight 

lines to the two foci, and pro- 

long the radius vector F'M till 

MH is equal to MF; the point 
. JZ belongs to the director circle 
\ described about F’ as a center ; 
' if a circle be described about 
/ M as a center with a radius 

equal to MF, this circle will be 
tangent to the director circle at 
H; on dropping from the focus 
Fig. 182. F a perpendicular upon this 

given line, and prolonging it till the line is double its origi- 





CHAP. VII. FOCI AND DIRECTRICES. 249 


nal length, a second point F, is found belonging to this same 
circle. The problem is reduced, therefore, to finding the center 
M of acircle passing through two given points F’ and F, and 
tangent to the director circle. For this purpose one constructs 
through the two given points / and F, any circle which inter- 
sects the director circle in two points #& and J'; draw from 
the point J, the intersection of the two lines FF, and KK’, a 
tangent to the director circle; the point M, where the line 
F'H intersects the given line, will be the point sought. 
One has, in fact, 


TH’ = 1K x IK'= IF x IF,; 


therefore, the circle which passes through the three points 
F, F,, H, is tangent to the director circle at H. Since two 
tangents can be drawn from the point J to the director circle, 
there will be two points M@ and M". 

When the point /, which is the symmetrique of the focus F 
with respect to the given straight line, is situated within the 
director circle, there are practically two solutions. In case 
the point F, is on the circle, the line is tangent to the ellipse. 
Finally, when the point F, is situated without the circle, the 
line does not intersect the ellipse. 


Foct AND DIRECTRICES OF THE HYPERBOLA. 


232. Since the equation of the hyperbola referred to its 
axes is 


it is sufficient to replace b? by — 0? in the results derived for 
the ellipse. One has then the two real solutions 

B=, e=tive’+P=H4+6, 
and the polynomial of the second degree is the square of the 


polynomial of the first degree a —_&. The remaining two 
solutions are imaginary. * 


a 


250 ; PLANE GEOMETRY. BOOK III. 


The hyperbola has therefore two real foci F and F’, situated 
on the transverse axis and at equal distances from the center 

. (Fig. 133). They are found by 
z’ RB drawing through the vertex A a 
/ straight line AG perpendicular 
to the transverse axis, meeting 
the asymptote in G, and laying 
off on the transverse axis the 
lengths OF and OF" equal to OG. 

The equation of the directrix 


2 
is «=~. The directrix DE 
a 

















Fig. 133. corresponds to the focus F, and 


the directrix D'E' to the focus F". Describe about the point O 
as center, with OA as radius, an are of a circle which inter- 
sects the asymptote in the point H; the point belongs to the 
directrix. The two triangles OAG, OHF, which have a com- 
mon angle O, the sides OA and OG respectively equal to OH 
and OF, are equal, and the angle OHF is right; if a perpen- 
dicular HD be dropped from the point H on the transverse 
axis OA, one has OH =OFx OD, and, consequently, 


2 
OD=". Thus the line DH is the directrix. 
Cc 


The constant ratio k= Vm?-+ n? is equal to ae this is the 
a 


eccentricity of the hyperbola; it is usually represented by the 
letter e. 4 


233. THueorem III. — The difference of the distances of each 
of the points of the hyperbola from the two foci is constant 
and equal to the transverse axis. — 

The distance from a focus to any point M of the curve is 


represented by + (« — ai The abscissas « and x of the focus 
a 


and of a point of the hyperbola being in absolute value greater 
than a, the second term is greater in absolute value than a. It 
is necessary, therefore, to give the — or + sign to the preced- 
ing parenthesis, according as this point M is on the right or 


CHAP. VII. FOCI AND DIRECTRICES. 251 


left branch, and as its distance is measured to the one or the 
other of the foci. In case of the right branch, one has 


MF=—a-+er, MF'=a+er; 
whence MF' — MF=2a. 
In case of the left branch 
MF=a—ex, MF'=—a— ex; 
whence MF — MF'=2a. 


234. Corottary I.— The difference of the distances of a 
point situated between the two branches of a hyperbola from the 
two foci is less than the transverse axis; in case the point is 
situated in either of the other two portions of the plane, the differ- 
ence is greater than the transverse axis. 

Let P be a point situated between the two branches of the 
curve (Fig. 134); the straight line 
PF meets the hyperbola at the point 





M. One has 4 N 
PF'— PM < MF’; 

if MF be subtracted from each mem- F’ F 

ber of the preceding inequality, it 


becomes 
PF' — PF < MF'— MF; 


this last difference is equal to 2a and 

therefore the first is less than 2a. Suppose now that P is 
situated to the right of the first branch of the hyperbola; the 
straight line NF" intersects this branch in M/; one has 


‘Fig. 134. 


NF < NM+ MP, 
and on adding to each member MF’, 
NF + MF'< NF'+ MF; 
whence NF'— NF > MF'— MF. 


The second difference being equal to 2a, the first is greater 
than 2 a. 


252 PLANE GEOMETRY. BOOK III. 


Whence it follows that the hyperbola may be considered as 
the locus of the points, such that the difference of their dis- 
tances from the two foci is equal to 2a. The construction of 
the hyperbola by points or by a continuous motion, given in 
the beginning (§ 14), depends upon this property. | 


235. CorotiAry II.— The distance of any point M of the 
hyperbola from the focus F' is equal to one of the normals drawn 
from this point to the circle de- 
scribed about the other focus F' as 
y center with a radius equal to the 

transverse axis. For a point M 


J < of the first branch (Fig. 135), one 
has | 
MF' — MF=2a= F'N, 
and, consequently, 
Fig. 135, MF = MF' — F'N= MN. 
For a point M' of the second branch, one has 
MEME =2a—) 1’, 
and, consequently, 
MEMES IEN HN. 
In the first case, the portion MN of the normal represents 
the distance of the point M from the circle, and the first 


branch of the hyperbola is the locus of points which are 
equally distant from the focus #’ and from the director circle. 








236. Tuzorem IV.— A tangent to a hyperbola bisects the 
angle formed by the radii vectores which are drawn from the 
point of contact to the foci. 

Let M and M' be two consecutive 
points on the hyperbola (Fig. 136). 
About the focus F’ as center describe 
with MF as radius the arc of a circle 
which intersects the radius vector FM' 
in C; about the focus F” as center 
describe an are of a circle with a 
radius equal to #'M which intersects 
the radius vector F’M’ in D; as the 








Fig. 136. 


CHAP. VII. FOCI AND DIRECTRICES. O58 


point M moves to the point M’, the two radii vectores receive 
the increments M'C and M'D; since the difference between 
the vectores is constant, these two increments are equal to one 
another. 

Lay off on the secant MM' an arbitrary length MG, and draw 
through the point G, GH parallel to the chord MC and Gir 
parallel to the chord MD. From this construction, it follows 

M'C _M'M_M'D. | 

MH M'G M'K’ 
since the two lengths M@'C and M'D are equal, it follows that 
the two lengths M'H and M'K are also equal. When the 
point M' approaches indefinitely the point M, the secant MM' 
approaches a limiting position and becomes the tangent to the | 
hyperbola at the point W@; at the same time, the chords MC 
and MD become tangents to the circles described about the foci 
as centers and, consequently, perpendicular to FM and F'M; 
the lines GH and GK, which are parallel to the chords, become 
also perpendicular to these same radii vectores, and the angles 
H and K become right angles. The two triangles M'GH, 
M'GK, which have acommon side M'G and a side M'H equal 
to M'K, become therefore right-angled, consequently equal to 
each other; whence it follows that the angles GM'H, GM'Ix 
become equal; thus the tangent to the hyperbola at the point 
M is the bisector of the angle FMF". 





237. CorotiAry I.— The hyperbola is the only curve which 
possesses this property; because on calling the radii vectores 
u and v, and their increments Aw and Av, one has 

Mie As 8 aes) OF 

Au M'C M'H 
If it be supposed that the angles GM'H, GM'K become equal 
when the point M' approaches indefinitely the point MW, the two 
triangles GM'H, GM'K become equal and also the sides M'H 
and M'iy; whence Ay 


lim — = 1. 
Au 





On returning to the primitive function, one has 
v=u+C, whence v—u=C. 


254 PLANE GEOMETRY. BOOK III. 


238. CoroLuary II.— An ellipse and a confocal hyperbola 
intersect at right angles. 

Two curves of the second degree are said to be confocal when 
their foci coincide; the angle at which two curves intersect is 
the angle formed by their tangents at the point of intersection. 
Let M be the point of intersection of 
an ellipse and a hyperbola which has 
the same foci F, F" (Fig. 137); the 
bisector MN of the angle MF" is, on 
the one hand, perpendicular to the 
ellipse, on the other, tangent to the 
hyperbola; therefore the tangents 
MT, MN to the curves are perpen- 
dicular to each other. 








Fig. 187. 


239. Proptem V.— To draw a tangent to a hyperbola at a 
given point M of the hyperbola. 

Take on the radius vector MF" a length MH equal to the 
other radius vector MF, and draw through the point © a line 
MP perpendicular to FH; one has the 
tangent required (Fig. 138). 

Remark. — It should be noticed that 
the tangent is wholly situated between 
the two branches of the hyperbola. 
Let P be any point of this tangent; 
one has 








PRS PH Pe, 
and, consequently, 
Fig. 138. PF'— PF<2a; 


therefore the point P lies between the two branches of the 
hyperbola. One branch of the hyperbola, lying always on the 
same side of any tangent to it, is a convex curve when viewed 
from any point of said tangents. 

The tangent being perpendicular to FH at its mid-point J, 
the point J is the projection of the focus / on the tangent. 
The straight line OJ, which is parallel to F’H and equal to the 
half of F'H, is constant; whence it follows that the locus of 


CHAP. VI. FOCI AND DIRECTRICES. 255 


the projections of the foci on the tangents is the circle 
described on the transverse axis as diameter. 


240. Prostem VI.— To draw a tangent to a hyperbola from 
any point P situated between its branches. 

Let PM be a tangent passing through the point P (Fig. 139). 

tf MH = MF be subtracted from the radius vector MF", one 
knows that the tangent 
PM is perpendicular to 
FH at its mid-point. The 
problem is reduced to de- 
termining the position of 
the point H; this point 
will be the intersection of 
the circle described: about 
the focus F" as center, with : 
a radius equal to 2a, and ee a 
the circle described about 9899 
the point P as center, with Fig. 189. 
a radius equal to PF. The tangent will be formed by drawing 
from the point P a perpendicular to FH, and the point of 
contact M will be determined by prolonging the radius vector 
F'H. These two circles intersect in a second point H'; a 
second tangent will be found by drawing a line through P per- 
pendicular to FH’; the point of contact of the tangent will be 
determined by the prolongation of the straight line F’Z'. 

In case the point P is on one of the asymptotes, one of the 
tangents drawn from P coincides with this asymptote, and 
- the point of contact is removed to infinity. 








241. Prostem VII.— To draw to a hyperbola a tangent 
which is parallel to a given straight line OL. 

One constructs about the focus F"’ as center, with a radius 
equal to 2 a, the director circle, and draws from the focus # a 
straight line perpendicular to OL (Fig. 140); this straight line 
intersects the circle in two points H and H'; straight lines 
are drawn through the mid-points of the lines FH and FH' 
parallel to OZ; these parallels will be the tangents required. 


256 PLANE GEOMETRY. BOOK IIl. 


The points of contact M and M' are determined by the lines 
Ths a Ae 3 i 

In order that the problem be 
possible, it is necessary that the 
given straight line, which can be 
assumed to be drawn through 
the center, does not intersect the 
hyperbola; for, then the perpen- 
dicular /'H' drawn from the focus 
F will intersect the director cirele 
in two points. } 








242. ProspiteM VIII.— To find the points of intersection of 
a straight line and of a hyperbola defined by its foci and its 
transverse axis. 

The construction is precisely the same as for the ellipse. 


THE Focus OF THE PARABOLA. 


243. The equation of the parabola, referred to its axis and 
to the tangent at the vertex, is 


y — 2px =0. 


Since this equation contains neither a term in xy nor one in 2”, 
one should have, according to the general relations of § 216, 
mn=0, 1 — m?=0; whence n=0, m=1. Because the coefficient 
of the term in y and the constant term are also zero, one has 
B=0, a —h?=0. Moreover, equations (5) of § 216 reduce to 


Po coee that is, a+h=p. The equation «?—h?=0 or 
P 





(a+ h)(a—h)=0 becomes pla —h)=0, that is, a—h=0; 
whence it follows that «=h= e 

Here one has asingle solution. Thus the parabola possesses 
a single focus situated on its axis at a distance from its ver- 
tex A equal to the half of its parameter (Fig. 141). The 
polynomial of the second degree being the square of the 


br 
an 
=! 


CHAP. VII. FOCI AND DIRECTRICES. 


po:ynomial of the first degree «+ 6 the dis- z 





ts 
y 
§ 


tance FM is equal to att. To the focus 


corresponds the directrix DE, whose equation , 





b> 


53 ; the directrix is perpendicular to 
the axis at a distance AD, equal to AF, from 
the vertex. 

The constant ratio 1 = Vm? + n* reduces in Fig. 141. 
this case to unity; whence it follows that 
every point of the parabola is equally distant from the focus 
and the directrix. 


is “2#=— 





244. TuroremM V.— The distance of any point within the 
parabola from the focus is less than its distance from the direc- 
trix; on the contrary, the distance of every point without from 
the directrix is less than its distance from the focus. 


Consider accordingly a point-N which les within the parab- 
ola; draw a perpendicular from this point to the directrix, 
and connect it with a straight line to the focus. The perpen- 
dicular intersects the curve in a point M which is joined to 
the focus. Since the point M belongs to the parabola, the 
distances ME and MF are equal. But the straight line NF 
is shorter than the broken line NM+ MF; if MF be replaced 
by its equal WE, it follows that the Teeter NF is less than 
NE. Thus the internal point WV is nearer to the focus than to 
the directrix. Consider next an external point P situated 
between the curve and the directrix. Connect it with the 
focus and draw to the directrix a perpendicular PE which is 
prolonged till it intersects the curve in M. Since the point 
M belongs to the parabola, the distances MF and ME are 
equal; the straight line M/F, or its equal ME, is shorter than 
the broken line MP + PF; if MP be subtracted from each 
member of the inequality, it follows that PE is shorter than 
PF. Incase the point P lies to the left of the directrix, it is 
evidently nearer to the directrix than to the focus. 

It follows from the preceding discussion that the parabola 
may be regarded as the locus of points, each of which is 


R 


258 PLANE GEOMETRY. BOOK IIt. 


equally distant from the focus and the directrix. It is in this 
manner that the parabola is defined in elementary geometry, 
and it is by means of this property that the parabola is con- 
structed point by point, or by a continuous motion. as has 
already been described (§ 16). 


245. THroremM VI.— The tangent to a parabola makes equal 
angles with the diameter and focal vector drawn from the point 
of contact. 


Take on the parabola two consecutive points M and ™’ 
(Fig. 142), which we join to the focus and from which we 
drop perpendiculars ME and M'E' upon the 
directrix. Construct an are MC of a circle, 
about the focus Ff’ as a center, with a radius 
FM, and draw from M, MC’ parallel to the 
directrix. The length M'C is the difference 
of the two radii vectores /'M' and F'M;; it is 
the increment which the radius vector 7M 
receives when the point M moves to M"'. 
Similarly, the length M'C' is the difference 
of the two perpendiculars M'E', ME; it is 
the increment which the perpendicular MH 
receives when the point Mis removed to M'’. Since the radius 
vector MF is always equal to the perpendicular ME, it follows 
that the two increments M'C and M'C' are equal. 

Draw through the points M and M’' the secant MS and con- 
struct the chord MC in the circle described about the focus as 
center. Take on the secant MS an arbitrary length MG, and 
draw through the point G, GH parallel to MC, and GIv par- 
allel to MC". On account of these parallels, one has the equal 
ratios aC MM _ MC. since the two lengths M'C, M’'C' 

MH MG Mk 
are equal, the two lengths M'H and M'K, which are propor- 
tional to them, are also equal. 

Suppose now that the point M’ approaches indefinitely the 
point M; the secant MS will approach a limiting position and 
be tangent to the parabola; the chord MC prolonged will, in a 
similar manner, approach a limiting position and be tangent 




















Fig. 142. 





CHAP. VII. FOCI AND DIRECTRICES. 259 


to the circle, and consequently become perpendicular to the 
radius FM; the parallel GH takes also a direction perpen- 
dicular to FM. Whence it follows that ; r 
the two triangles M'GH, M'GK have a 
as limits the two right-angled triangles i oe L 
MGH, MGK (Fig. 143); these two right 
- triangles, having the common hypotenuse i tm--ty 
MG, and the two sides MH and MK F 
equal to each other since they are the 
limits of equal lengths, are equal; hence 
the two angles GMA and GMH are 
equal. Therefore the tangent M7’ to the 
parabola bisects the angle FME, formed 
by the radius vector and the perpendicular dropped from the 
point of contact to the directrix. If EM be prolonged, the 
two vertical angles GMA and T'MZ will be equal and, conse- 
quently, the two angles FMT, T'ML, formed by the tangent 
with a line parallel to the axis and the radius vector FM, are 
equal. 











Fig. 143. 


246. CoroLttARy JI.—Suppose that a light be placed at the 
focus F (Fig. 144) of the parabola; the rays of light, emanat- 
ing from the focus F, are reflected on meeting the parabola, 
making the angle of reflection equal to the 
angle of incidence. Let FM be one of the 
rays; draw at the point M a tangent to 
the parabola; the reflected ray, making the 
angle LMT" equal to the angle FMT, will be 
parallel to the axis AB of the parabola. Simi- 
larly, every reflected ray will be parallel to 
the axis. 

It is by means of this property of the 
parabola that the reflectors used in reflector 
telescopes and coach lamps are constructed. 
The interior surface, of well-polished metal, is produced by a 
parabola revolving about its axis; a light is placed at the 
focus; the luminous rays, after reflection, all become parallel 
to the axis; the reflector projects a pencil of parallel rays 








Fig. 144. 


260 PLANE GEOMETRY. BOOK III. 


which are propagated without dispersing, and which light, 
therefore, at the greatest distance. 

CoroLtuAry II.—Suppose that luminous rays, parallel to 
the axis, fall upon a parabolic mirror; after reflection, they 
will all converge to the focus. 

Parabolic mirrors are used in the construction of telescopes. 
The axis is directed toward the star; the luminous rays coming 
from the star are reflected on the mirror, and form at the focus 
a very brillant image of the star. 

The parabolic form is used in the construction of speaking- 
trumpets and certain acoustical instruments. 


247. CorotuAry III. — Conversely, the parabola is the only 
curve which enjoys this property that the tangent at any point 
of the curve makes equal angles with the radius vector drawn 
from a fixed point to the point of contact, and with a straight 
line drawn from the point of contact parallel to a fixed straight 
line. Imagine that any point MW of the plane be determined by 
its distance M/F from a fixed point F, and its distance ME 
from a straight line DE perpendicular to the fixed straight 
line FB (Fig. 142); represent these two co-ordinates by u and 
v (§ 17). As any point WM of the curve is moved to a neigh- 
boring point M’, these two co-ordinates receive the increments 
At= MC, A= WC’, 

AD MG ae 

Au MOC MH C4 
As the point ™M’ approaches indefinitely the point MM, the 
straight line MM' becomes tangent and the angle at H becomes 
a right angle. The two triangles GMH, GMK are at the 
limit right-angled and equal (Fig. 143), since they have a 
common hypotenuse and the angle GMH equal to GMI by 
hypothesis. Therefore one has 


and one has 





Re 
lm as if 


and returning to the primitive function y=uw+C. On remoy- 


ing the line DE a distance equal to the constant C, it follows 
that v = u. 


CHAP. VII. - FOCI AND DIRECTRICES. 261 


248. Prosiem 1X.— To draw a tangent at a given point of a 
parabola. 


First Meruop.— Let 7 (Fig. 145) be the point in which 
the tangent prolonged intersects the axis, ME the perpen- 
dicular drawn from the point M to the directrix. It is known 
that the tangent bisects the angle FME; the angle FTM being 
equal to the alternate interior angle TME, and, consequently, 
to the angle FMT, it follows that the 
triangle TFM is isosceles, and the two 
sides FM, FT are equal. Hence, in order 
to construct the tangent at the point ™, 
it is sufficient to lay off on the axis a 
length F7' equal to the radius vector /'M, 
and draw TM. This method is not prac- : 
tical in case the point M is very near the : 
vertex A of the parabola; for then the 
two points M and 7, being very near to 
each other, do not determine the tangent with sufficient 
precision. For this particular case the following method is 
used. 

















Fig. 145. 


Suconp Mrernop.— The tangent MT bisects the angle at 
the vertex M of the isosceles triangle FME, and is perpen- 
dicular to the base FE at its mid-point. Thus, in order to 
construct the tangent a perpendicular WE is drawn from the 
point M to the directrix, and a second perpendicular is drawn 
from the point M to the straight line FE. 

It follows from this construction that the tangent at the 
vertex A of the parabola is perpendicular to the axis of the 
parabola. 


Remark. — Every point of the tangent, excepting the point 
of contact M, lies without the parabola. Let P be any point 
of the tangent; it is perpendicular to FE at its mid-point ; 
therefore, the distances PE, PF are equal; but the oblique 
line PE is greater than the perpendicular PX; therefore, the 
distance PF is greater than PK, and, consequently, the point P 
is without the parabola. Whence it follows that the parabola 
is a convex curve when viewed from any point on a tangent. 


2962 PLANE GEOMETRY. BOOK III, 


249. CoroLtiary. — The locus of the projections of the focus 
upon tangents to the parabola is the tangent at the vertex. 


In fact, it is seen that the point J, mid-point of FE, and the 
projection of the focus upon the tangent, lie on a parallel to 
the directrix drawn through the point A, the mid-point of FD, 
that is, on the tangent at the vertex A. 


250. Prostem X.— To draw a tangent from an external 
point P to a parabola. 

Assume that the problem is solved, and let PM (Fig. 146) 
be a tangent passing through the point P. If a perpendicular 
ME be drawn from the point M to the directrix, and the 
points # and F be joined, it follows that 
the tangent PM is perpendicular to FE at 
its mid-point; whence it follows that the 
distance PE is equal to PF, and one has 
the construction required: a circle is 
described about P as a center, having a 
radius equal to the distance PF of this 
point from the focus and intersecting the 
directrix in the point £. Join the points 

Fig.115. F and E, and draw from Pa perpendicular 
to FE; it will be the tangent required. The point of contact 
M is determined by the intersection of the tangent with the 
line drawn through the point EZ parallel to the axis, 

The circle intersects the directrix in a second point EZ’. In 
a similar manner a perpendicular is drawn from the point P to 
FE’, and a second tangent is constructed. 

These constructions can be accomplished without tracing the 
parabola. It is only necessary that the focus and directrix be 
known. 











251. Prostem XI.— To draw to a parabola a tangent which 
is parallel to a given straight line KL. 

Assume that the problem is solved, and let M7’ be the 
tangent required. If a perpendicular ME be drawn from 
the point of contact to the directrix, and the points # and F 


CHAP. VII. FOCI AND DIRECTRICES. 263 


be joined, then the tangent is perpendicular to FE at its 
mid-point. | 

Whence the following construction 1s 
deducible: Draw through the focus F a jy -------- x 


straight line perpendicular to the given | 
a 


line KD, and produce it till it meets the 
directrix in Z, and at the mid-point of FE ‘ 

erect a perpendicular TM, which will be 

the tangent required. The point of con- }“« 

tact M will be determined by drawing 

through the point E the line ME parallel 

to the axis. Fig. 147. 





, 
oe 








252. Proptem XII.— To find the point of intersection of @ 
given straight line and of a parabola defined by its focus and 
directrix. 

_ Let the point F, be a point which is symmetrical to the focus 
with respect to the given line (Fig. 148). 
The point M, being equally distant from the 
points 7’, F,, and the directrix, is the center 
of a circle passing through these two points 
and tangent to the directrix. In order to 
determine the point of contact H, one lays 
off on the directrix, beginning at the point [ 
in which the straight line FF intersects the 
directrix, to the one side or to the other, a 
length JH which is a mean proportional 
between the two lengths IF, JF,; thus are is we 
the two points of intersection M and M' determined. 

-In case the point F,, the symmetrique of the focus with 
respect to the given line, is situated to the right of the directrix, 
there are two solutions. When the point Ff is on the directrix, 
the line is tangent to the parabola. Finally, when the point 
F, lies to the left of the directrix, the straight line cannot 
intersect the parabola. 








264 PLANE GEOMETRY. BOOK III. 


253. THrorem VII. — The limiting case of an ellipse or of a 
hyperbola whose parameter remains finite, while the major or 
minor axis increases indefinitely, is a parabola. 

The ordinate at the focus in 
the parabola is equal to the 
parameter p; by analogy, the 
ordinate at the focus in the 
ellipse and the hyperbola is 
called the parameter ; itis equal 


> 2 
A = 


to” and is represented by p. 
a 





The ellipse, referred to its 
major axis and to the tangent 
at the left vertex (Fig. 149), 
Fig. 149. has an equation of the form 








D p2 2 
a es or y= 2px —Py?, 
a a a 


Assume now that the vertex remains fixed, and the parameter 
p remains finite, while the major axis 2a is allowed to increase 
indefinitely ; the equation of the ellipse is reduced to the equa- 
tion y= 2px, which represents a parabola. If the points, 
which correspond to the same value of 2, be considered, one 
sees that each point of the parabola is the limiting position 
toward which the corresponding point of the ellipse tends when 
a is increased indefinitely ; it is this that is implied in saying 
that the parabola is the limit of the ellipse. 

The equation of the hyperbola, referred to its major axis and 
to the tangent at the vertex A, is 


y? = 2 px + Ea, 


if a be allowed to increase indefinitely, the parameter p remain- 
ing finite, this equation will also reduce to 


y ome 2 px. 


CHAP. VII. FOCI AND DIRECTRICES. 265 


The parabola is the limit of the branch of the hyperbola to 
which the vertex A belongs; the other branch is removed 
indefinitely toward the left. 

In the preceding discussion we have supposed that the 
parameter of the ellipse or the hyperbola remains finite. The 
same conclusion is reached, on supposing that the distance 4F’ 
of the vertex A from the neighboring focus F’ remains finite. 
In fact, on calling « this distance, one has, for the ellipse, 


2 Ver; os. 
eee cf _@ Met =o 2-5); 
a a 





since the parameter p has as limit the finite quantity 2, the 
equation of the ellipse reduces to y2=4 aa. The same will be 
the case for the hyperbola. 


254. Remark. — This transformation of the ellipse into the 
parabola is important. It allows the deductions of the proper- 
ties of the parabola from those of the ellipse as particular cases. 
Thus, in the ellipse, the diameter, or the locus of a system of 
parallel chords, is a straight line passing through the center; 
if it be supposed that the center is removed to infinity, the 
ellipse is transformed into the parabola, and the diameters 
become parallel to the axis. The ellipse is the locus of points 
equally distant from the focus # and from the director circle 
described about the focus F” as center (§ 221). If the focus F" 
be removed to infinity, the director circle becomes the directrix 
of the parabola. 

The tangent to the ellipse makes equal angles with the radu 
vectores drawn from the point of contact to the foci (§ 222); 
if the focus F’ be removed to infinity, the radius vector MF 
becomes parallel to the axis. 


255. Tueorem VIII. — Jf two tangents be drawn to a curve 
of the second degree, the straight line FP, which is drawn from 
the focus F' to the point of intersection P of the two tangents, is 
the bisector of the angle formed by the radii vectores FM, FM", 
drawn from F to the points of contact of the tangents, or the 


266 PLANE GEOMETRY. BOOK III. 


external angle, according as the two tangents touch the same 
branch of the curve or two different branches. 

Consider two tangents PM, PM' of an ellipse (Fig. 150); 
prolong the radius vector #'M till 
MH is equal to MF, and similarly 
FM' till M'H' is equal to M'F’; 
the tangents being perpendicular at 
the mid-points of FH and FH", it 
follows that 


PH= PF, PH'=PF', 





Fig. 150. and the two triangles #'’PH, H'PF 
are equal, since the three sides of 
the one are equal each to each to the three sides of the other, 


namely, 
F'H = FH'=2a, PH= PF, PF'= PH'; 


whence it follows that the angles PHM, PF'M' are equal. But 
the angle PHM is equal to the angle PIM, therefore the 
angles PFM, PFM' are equal and the straight line FP is the 
bisector of the angle MF'M". 

The same discussion holds in case the locus is a hyperbola, 
when the two tangents touch the same branch; but in case 
the tangents touch two different branches, the line FP is the 
bisector of the angle formed by one of the radii vectores PM 
and the prolongation of the other. a 

Consider, finally, the case when the curve is a parabola (Fig. 
151). From the points of contact draw the perpendiculars 
MH, M'H' to the directrix; since the tangents are perpen- 
dicular to FH and FH' at their mid-points, the angles PF'M, 
PFM' are equal respectively to the angles PHM, PH'M'. The 
straight lines PH and PH', being each equal to the straight 
line PF, are equal to each other, and the triangle HPH' is 
isosceles. The angles PHM, PH'M', complements of the 
equal angles of the isosceles triangle, are equal to each other; 
therefore the angles PFM, PFM' are equal. This result may 
otherwise be obtained immediately on regarding the parabola 
as the limiting case of an ellipse. 


CHAP. VII FOCI AND DIRECTRICES. 267 


256. TurorEeM IX. — Tangents drawn from an exterior point 
P to an ellipse or a hyperbola, make equal angles with the straight 
lines drawn from this point to the foci. 

In the two equal triangles /’PH, H'PF (Fig. 150), one has 
the two equal angles F’PH, H'PF; on subtracting the com- 
mon part F'PF, one has FPH = F'PH', and, on taking half 
of the remainders, one obtains PM = F'PM". 

The same property belongs to the parabola, considered as 
the limiting case of an ellipse; it suffices to replace the radius 
vector PF' by a straight line PI 
parallel to the axis (Fig. 151). It is 
easy, moreover, to demonstrate this 
property directly. Ifabout the point / 
P as a center, a circle be described 
with a radius equal to PF, this circle | 
will pass through the points H and 
H'; the angles MPI, FHH' are equal, 
since their sides are respectively per- 
pendicular; but the inscribed angle 
FHH' is the half of the angle FPH' 
at the center, and, consequently, 
equal to the angle /PM'; therefore the angles MPI, M'PF 
are equal. 














Fig. 151. 


257. TueorEM X.— The straight line F'K, which joins the 
focus of a curve of the second degree with the point in which any 
secant intersects the directrix, is the bisector of the external angle 
formed by the radii vectores emanating from the focus to the 
points in which the secant cuts the curve or bisector of the angle 
included by the same radii vectores, according as the two points 
of intersection, M and M', are situated on the same branch, or 
different branches of the curve. 

- Draw from the points M and M' perpendiculars to the direc- 
trix (Fig. 152); one has 
MF _ MF 
ME M'E” 
MF ME MK 


and, consequently, MF ME MK 








268 PLANE GEOMETRY. BOOK III. 


In case the two points M and M' belong to the same branch 

oe of the curve, since the point A: lies 
: _on the prolongation of the chord 
, MM’, the straight line FA is the 
* bisector of the external angle of the 
triangle MF'M'. In case the points 
M and M’' belong to two different 
branches, since the point £ is situ- 

Fig. 152. ated between the points M and M’, 
the straight line F'X is the bisector of the angle MFM". 








258. THrorEM XI.— Jf tangents be drawn from any point 
P on the directrix to a curve of the second degree, the chord of 
contact MM' passes through the cor- 
responding focus F, and is perpendic- 
ular to the straight line FP which joins 
the point P to the focus (Fig. 153). 

Let the tangent PM be the limiting 
position of a secant of the ellipse 
whose points of intersection with the 
ellipse are made to coincide; then it 
follows from the preceding theorem 
that the line FP is perpendicular to FM; it is for the same 
reason perpendicular to /-M'; therefore the line MFM’ will be 
a straight line perpendicular to FP. 


259. THrorem XII.— The product of the distances of the 
two foci from the tangent of an ellipse or a hyperbola is constant. 
Let FH, F'H' be the perpendiculars dropped from the foci 
upon a first tangent (Fig. 154), FX, F'K' the perpendiculars 
dropped upon a second tangent, P the 
point of intersection of the two tan- 
gents. Then by Theorem IX. it fol- 
lows that the right triangles PH, 
F'PkK' are similar, so also are the 
triangles FPK, F'PH', and one has 
ie he ee 
PK FP FH’ 
Fig. 154. whence FH. F'H'=FK. F'K'. 








Fig. 153. 





CHAP. VII. FOCI AND DIRECTRICES. 269 


If the curve be an ellipse, in drawing the tangent parallel to 
the major axis, it follows that the constant product is equal 
to b2. When the curve is a hyperbola, if the asymptotes be 
regarded as the limiting position of tangents, one sees also 
that the product is equal to 0’. 


260. Propiem XIII.— To construct a curve of the second 
degree, given the focus F and three points A, B, C. 


Assume that the problem is solved and that the three points 
belong to the same branch; the point D, where the secant AB 
is met by the bisector of the exterior angle of the triangle 
AFB, is on the directrix (§ 257); the 
secant BC will determine in a simi- 
lar manner a second point D!' on the 
directrix. The focus F, the direc- 
trix DD', and the point A define a 
curve of the second degree and one 
only; it will be an ellipse, a parab- 
ola, or a hyperbola, according as 
the distance AF is less, equal to, or 
greater than the distance AZ of the point A from the direc- 
trix. It is easily seen that this curve passes through the two 
points B and C; for, on account of the bisector FD, one has 


AF _AD_ AE 
BF BD BE? 








Fig. 155. 





and, consequently, an = a 


therefore the curve passes through the point B. It can be 
shown ina similar manner that the curve passes through C. 
This gives one solution. 

It is possible that the three points are not on the same 
branch; if, for example, the two points, A and B, are on the 
same branch and the point OC on the other branch of the 
hyperbola, the bisectors of the angles AFC, BFC will deter- 
mine two points on the directrix. The three solutions found 
in this manner are hyperbolas. One has therefore, in all, four 
solutions; of these four curves of the second degree to which 


270 PLANE GEOMETRY. BOOK III. 


the given focus belongs and on which the three given points 

lie, three are always hyperbolas, the fourth is an ellipse, a 
hyperbola, or a parabola depending upon the disposition of the 
points. 


261. A calculation will lead to the same result; let @ and 
B be the co-ordinates of the focus, wv’ and y’, #" and y!, a!!! 
and y'", the co-ordinates of the three given points, 8’, 8", 8!", 
their distances from the focus; the equation of the curve can 
be put under the form 


(a — a)? +(y— BY? — (ma+ny+hy=0, 


where ma + ny + h=0, is the equation of the directrix. One 
can determine the three constants m, n, h by means of the 
three equations of the first degree: 


6’ =+(me' +ny' +h), 
8” =+ (ma! + ny" +h), 
pee (mar! + ny!" + h). 


Each combination of signs furnishes a system of equations; 
there are eight combinations; but it is to be noticed that, if 
the signs be changed in the three equations, the values of m, 
n, h change signs, and the curve is the same; therefore there 
are only four solutions. 

The distance of a point from a straight line is expressed by 
a formula affected with a double sign; the same sign should 
be taken for any point lying on one side and the opposite sign 
for any point situated on the other side of the line. One 
knows that the ellipse lies wholly on the same side with re- 
spect to each directrix; the parabola is also situated on the 
same side of its directrix, but, however, the two directrices 
of the hyperbola le between the two branches of the curve. 
When the three points lie on the same branch, their distances 
from either directrix have the same sign; in case, however, 
two of the points lie on one branch and the third on the other 
branch, these distances take different signs. 


CHAP. VII. FOCI AND DIRECTRICES. Prat 


262. Propirem XIV. — Construct a curve of the second degree, 
when one focus and three tangents are given. 


Assume that the problem is solved; if perpendiculars be 
dropped from the given focus upon the given tangents, and 
~ each prolonged a length equal to itself, three points, H, H’, 
H", are determined, belonging to the director circle (Fig. 156) 
whose center is at the second focus F"; 
the radius F"H of this circle is equal to 
the axis 2a which passes through the 
two foci. The two foci F, F", along with 
the length 2a, define a curve of the sec- 
ond degree, and one only. It is easy 
to see that this curve is tangent to the 
three given lines, for let M be the 
point in which the line #"# intersects 
the straight line MT, the sum or the 
difference of the radii vectores MF" and Fig. 156. 

MF being equal to F'H or to 2a, the 

point M belongs to the curve; further, the straight line M7, 
being perpendicular to FH at its mid-point, is tangent to the 
curve at the point M@ The problem has thus one, and one 
solution only. 

If the three points H, H', H' should lie on the same straight 
line, the curve sought would be a parabola having this line for 
its directrix. 





TRINOMIAL EQUATION COMMON TO THE THREE CURVES 
OF THE SECOND DEGREE. 


263. If a point O of a curve of the second degree be taken 
for the origin, the diameter meeting the curve in this point 
for the x-axis, and the tangent at this point for the y-axis, the 
equation of the curve takes the form 


y? = 2 px + qa’. 
In fact, let 


Av? + 2 Bey + Cy? +2 De+2Hy+F=0 


212 PLANE GEOMETRY. BOOK IIT. 


be the equation of the curve referred to the axes mentioned. 
Since the curve passes through the origin and is tangent to 
the axis Oy, one has roche peed || 

cor ese ) 


since the axis Ox is the diameter conjugate to the chords 
parallel to the axis Oy, the equation should contain only the 
second power of y, because to each value of # there should 
correspond two equal values of y with opposite signs; there- 
fore B=0. The coefficient C is not zero, because if it were, 
the conic would reduce to two straight lines parallel to the 
axis Oy. Hence one can solve the equation with respect to 
y’ and obtain an equation of the form given above. The curve 
is an ellipse, a hyperbola, or a parabola, according as g 1s nega- 
tive, positive, or zero. 

Take, in particular, the point in which the focal axis meets 
the curve for the origin, and the direction in which one looks 
from this point toward the nearest focus for the positive 
direction of the axis Ox. Whence the coefficients p and gq will 
have the following values: 

1° Ellipse. Oncalling a and b the axes of the ellipse, one 
ought to have y=0 for x=2a, and y°=6? for <=a. Therefore 

pa+qa’?=0, 2pat+gqa’=6*; 
b? ele 3 


whence p=— G=->5=5-- 
a ee 


1=e?—1. 


2° Hyperbola. One should have y=0 for «=— 2a, and 
y?=—b*? forx=—a. Therefore Z 


—pa+qv’=0, —2patgqa?=—?d 


whence p=-, q=—5=,4—1=-e-1. 
a | 

3° Parabola. Here q is equal to zero, and p is the parameter. 

In general, therefore, on taking the point in which the 
straight line drawn through the foci intexsects the curve for | 
the origin, and the straight line drawn from this point to 
the nearest focus for the a-axis, one can put the equation of 
the three curves under the form 


y? = 2 pa +(e? — 1) 2’, 


CHAP. VII. FOCI AND DIRECTRICES. 273 


in which p is the parameter and e the eccentricity, which is 
greater than unity for the hyperbola, less than unity for the 
ellipse, and equal to unity for the parabola. 


THE EQUATIONS OF THE CURVES OF THE SECOND 
DEGREE IN POLAR CO-ORDINATES. 


264. A focus F’ is chosen as the pole, and the perpendicular 
drawn from this focus to the corresponding directrix DE 
is taken for the polar axis. 

Consider now the ellipse. The ratio of the distances of any 
point M of the curve to the 
focus and to the directrix 


being constant and equal to M i 
the eccentricity, one has 
eg 





1 or MF= ME -e. 
ME 


The distance FD of the 


focus to the directrix is equal Fig. 157. 
2 


to”. On projecting the broken line FME (Fig. 157) upon 
c 





the axis, one has 
2 
pcosw+ME=FD=", w= Z XFM, 


Ub? 
whence ME = eT PCOS w; 


on replacing ME by its value | 

in the preceding equation, one ve | 
finds M \E m/ 

Oo, Sane 


P 1+ e cosa F a] |[D A’ Fae 








If the curve be a hyperbola 
- (Fig. 158), the same calculation 
is applicable to the branch A, 
whose vertex is nearer the 
focus # taken for the pole. When the point M' is on the 


i) 





Fig. 158. 


274 PLANE GEOMETRY. BOOK III. 


branch A’, situated on the other side of the directrix, the pro- 
jection of the broken line F'M'E gives 


p COs wo — ME = FD, 


which leads to the equation 





ae 
2) Pe 1 —ecosw 
Moreover, if the negative radii vectores be constructed in a 
sense contrary to the direction indicated by the angle o, it 
is easily seen that equation (1) represents the two branches 
of the hyperbola. Let M' be any point of the second branch, 
w' the corresponding angle A'F'M', p' the radius vector FM’; 


* : oe —p : 
owing to equation (2), one. has p'= reer If in equa 
tion (1), the value w'+ 7 be substituted for the angle o, it 


will become 





Mio. 7 
P 1 — e tos wo! is 





Thus a negative value —p! is obtained for p. But the value 
w +7 assigned to w indicates the direction FM, opposite to 
FM'; if p have a positive value, it will be necessary to measure 
it in the direction FM; p having a negative value — p', one 
measures the absolute value p!' in the opposite direction; that 
is, in the direction F'M', which determines the point J". 
Whence it follows that equation (1) suffices to represent the 
two branches of the hyperbola, the first 
by the positive values of p, the second by 
the negative values. 
ae The calculation given for the ellipse 
is applicable to the parabola (Fig. 159); 
it suffices to put e=1. It follows there- 
fore that equation (1) represents the 
three curves of the second degree; the 
curve is an ellipse, a parabola, or a 
hyperbola according as the eccentricity 
e is less, equal to, or greater than unity. 


f 








Fig. 159. 


CHAP. VII. FQCI AND DIRECTRICES. 275 


EXERCISES. 


1. If P be the point of intersection of the tangents drawn to 
a parabola at the points Mand M'and F the focus, prove that 


Pat Fae 
MF M'F 

2. In case of a curve of the second degree, show that the 
perpendicular dropped from the focus upon a chord and the 
diameter conjugate to this chord intersect on the directrix. 

3. A semi-diameter of an ellipse or of a hyperbola is a mean 
proportional between the straight lines which join the foci to 
the extremity of the diameter conjugate to the first. 

4. Show that the distance of any point of an equilateral 
hyperbola from its center is a mean proportional between the 
distances of this point from the foci. 

5. Find in the plane of an ellipse a circle such that the 
length of the tangent drawn from every point of the ellipse to 
the circle is a rational, integral function of the first degree in 
the co-ordinates of this point. | 

Prove that the sum or the difference of the tangents drawn 
from every point of the ellipse to two circles which enjoy the 
preceding property is constant. 

6. Find the locus of the vertex of a constant angle which is 
circumscribed about a parabola. 

7. A chord is drawn through the focus of a parabola, and a 
circle is constructed on this chord as a diameter, then tangents 
are drawn to the circle parallel to a given straight line; find 
the locus of the points of contact. 

8. A constant angle revolves about the focus of a curve of 
the second degree; tangents are drawn to the curve at the 
points in which the sides of the angle meet this curve; find 
the locus of the point of intersection of the tangents. 

9. A tangent is drawn to a given ellipse at any point M and 
is prolonged till it intersects the tangents at the extremities of 
the major axis in P and Q; find the point of intersection N of 
the straight lines F’P and FQ, and of the point of intersection 
N' of the straight lines FP and F’Q. Show that the two 
points NV and NW’ are situated on the normal at the point M. 





276 PLANE GEOMETRY. BOOK Itt. 


10. A curve of the second degree is given, and a secant re- 
volves about a fixed point P; the focus F is joined to the 
points M and M'in which the secant intersects the curve; 


! 
PFM ,,,, PFM 








show that the product tan is constant. 


11. Show that the portion of a tangent comprised between 
two fixed tangents to a curve of the second degree subtends 
a constant angle whose vertex is at a focus of this curve. 

12. Prove that the point of intersection of the altitudes of a 
triangle circumscribed about a parabola is on the directrix, and 
that the circle circumscribed about the triangle passes through 
the focus. 

13. If, at any point M of an ellipse, a normal be drawn, the 
portion of this normal comprised between the point M and the 
minor axis has for its projection on the radii vectores drawn 
from the point M to the two foci a length equal to the semi- 
major axis. 

14. Prove that the portion of the normal comprised between 
the point M and the major axis has for its projection on the 
radii vectores a length equal to the parameter of the ellipse. 

15. Two curves of the second degree have a common focus ; 
if radii vectores be drawn from this focus to the extremities 
of any diameter of one of the curves, the sum or the difference 
of the ratios of these radii vectores to the radii vectores of the 
second curve, which have the same direction, is constant. 

16. If the radii vectores which are drawn from any point M 
of an ellipse be prolonged till they intersect the curve at P and 
Q, show that the sum ae is constant. 

17. A mariner’s compass composed of m rays revolves about 
its center placed at the focus of an ellipse; show that the 
sum of the inverse of the lengths intercepted on each ray be- 
tween the focus and the point where it intersects the ellipse, is 
constant. 

18. From any point P situated in the plane of an ellipse, 
tangents are drawn to this ellipse; a perpendicular PC is 
dropped from the point P to the chord of contact AB; the 
straight lines PC and AB intersect the minor axis in Dand £; 


CHAP. VII. FOCI AND DIRECTRICES. yas 


show that the circle described on DE as a diameter passes 
through the two foci. 

19. Being given two confocal ellipses, through a point P one | 
draws to one of them tangents which intersect the second, the 
one in A and B, the other in Cand D; demonstrate that 


1 1 1 a 

PA* PB PO~ PD 

20. A circle is described on the major axis of an ellipse as 

a diameter; the ordinate of any point M of the ellipse inter- 
sects the circle in a point N; if » be the angle which the 
radius vector FM makes with the major axis, and U the angle 
which the radius vector ON of the circle makes with the 
major axis, one has the relations 








o .j/l+e uw 

Zee Nieee ae 

On representing the area of the elliptic sector AFM by S, one 
has also 


p=a(1—ecosu), tan 


S=Lu-e sin wv). 


21. An equilateral hyperbola confocal to an ellipse inter- 
cepts, on the sides of a right angle circumscribed about an 
ellipse, two equal chords. 

22. If one call R the radius of the circle circumscribed 
about a triangle which is inscribed in a parabola, ¢, ¢', c’’ the 
chords drawn from the focus parallel to the sides of the tri- 
angle, 6, 6', 6" the angles which the sides of the ee make 
with the axis, one has 

Rsin6'-sin6"=p, 8pR?=ce'c". 

23. Let A be the vertex, / the focus of a parabola, (p, w), 
(p', w') the co-ordinates of two points M and M' of the curve, 
6 the angle MFM', S the area of the sector AFM, A that of 
the sector MFM", 1 the length of the chord MM'; demonstrate 
the following formulas used in astronomy : 


2 pp! sin®S 
So GONE: +008 $ = Oe 
p +p’ 2V pp! C08 5 








278 PLANE GEOMETRY. BOOK III. 


=<F(p + p)Vp(2p—p —p=15 tan 3(1 + tan? >) 


sve sham (e+e +) pp! cos 5) 


ae + p'+-V pp! cos 5) y20(0 + p!—2-V pp! cos >) 


_V2pl (ote! t+? /pt+e'—)\F] 
= 6 7) 2 


24. Consider an ellipse referred to its two axes, and an ex- 
ternal point P whose co-ordinates are a and B; two tangents 
are drawn from the point P to the ellipse, and each of the 
points of contact are joined to the two foci. Demonstrate: 
1. That the distances of the point P from any of the four 











lines thus constructed is equal to : V a?B? + ba? — a*b?, a and b 
a 


representing the semi-axes of the curve; 2. That the sum or 
the difference of the tangents drawn from the two foci to the 
circumference of the circle described about P as a center, with 
a radius S, is equal to 2a. 
25. Calculate the parameter and eccentricity of a curve of 
the second degree, defined by its general equation (use the for- 
mulas of § 143). 


CHAP. VIII. THE CONIC SECTIONS. 279 


CHAPTER Vit 
THE CONIC SECTIONS. 


265. Turorem I.— The section of a right circular cylinder 
made by any plane oblique to the base 1s an ellipse. 


Draw through the axis OO' of the cylinder (Fig. 160) a 
plane perpendicular to the secant plane; this plane is taken 
as the plane of the figure. The 
plane intersects the cylinder in 
two diametrically opposite gen- 
eratrices GG', HH', and the se- 
cant plane in the straight line 
AA'. Describe in the plane of 
the figure two circles O and O' 
tangent to the line 4A’ and the 
two generatrices GG', HH’ of 
the cylinder; draw the _bisec- 
tors of the angles A and A' and 
produce them till they inter- eee 
sect the axis of the cylinder in eae 
O and O'; if, about the point O as a center with the radius of 
the cylinder, a circle be described, this circle will touch the 
generatrices in G and H, and the straight line AA! at the point 
F; the circle described about the point O' as center will in 
a similar manner touch the generatrices in G' and H' and the 
straight line 4A’ at the point #”. Imagine that the figure 
be revolved about the axis OO'; the generatrix GG' will 
generate the surface of the cylinder while the two circles will 
generate two spheres inscribed in the cylinder and touching 
it internally, the first along the circumference of the great 
circle GLH, the second along the circumference of the great 
circle G'L'H'. Moreover, the two spheres are tangents to the 














280 PLANE GEOMETRY. BOOK III. 


given plane, the first at the point F, the second at the point 
fF". In fact, the plane of the figure and the given plane are 
perpendicular to each other; the straight line OF, which lies 
in the first plane and is perpendicular to their intersection, 
is perpendicular to the second plane; the plane AMA’ being 
perpendicular to OF at its extremity, is tangent to the sphere 
O at the point F. It may be shown in a similar manner that 
the given plane is tangent to the sphere O' at the point 7”. 

Let AMA' be the curve in which the secant plane intersects 
the cylinder; it will be demonstrated that this curve is an 
ellipse whose foci are the points F and F’. Join any point M 
of this curve to the two points F and F’ and draw a generatrix 
of the cylinder through the point M; this generatrix is tangent 
to the upper sphere at the point Z, and the lower sphere at the 
point L'. The two straight lines MF, ML, tangents drawn 
from the same point M to the sphere O, are equal; similarly, 
the two straight lines MF’, ML', tangents drawn from the 
point M to the lower sphere, are equal. Hence the sum of the 
radii vectores MF'+ MF' is equal to ML + ML’, that is, equal 
to the portion LL’ of the generatrix comprised between the 
two circles of contact; this length is constant, because, by the 
revolution of the figure about the axis OO’, the generatrix GG! 
is made to coincide with ZL’. Hence it follows that the sum 
of the distances of each of the points of the curve from the two 
fixed points F' and F’ is constant and equal to GG", and conse- 
quently that the curve is an ellipse whose foci are Fland F". 

Corotiary. — The straight lines DE and D'E', the inter- 
sections of the secant plane and the planes of the circles GH 
and GH", along which the inscribed spheres touch the cylinder, 
are the directrices of the ellipse. In fact, if a plane be drawn 
through the point M perpendicular to the axis of the cylinder, 
the section of the cylinder by this plane will be acircle NMN’. 
The straight line DE, the intersection of the two planes, which 
are perpendicular to the plane of the figure, is also perpendicu- 
lar to this plane and, therefore, to the straight line AA'; in 
the same manner it follows that the straight line MP, the in- 
tersection of the plane of the circle and of the secant plane, is 
perpendicular to AA’. Since the radius vector MF is equal to 


CHAP. VIII. THE CONIC SECTIONS. 281 


ML or to NG, and the perpendicular dropped from the point 
‘M upon the directrix DE is equal to PD, the ratio of the dis- 
tances of the point M from the focus and to the directrix 1s 


ae. but since PN and GD are parallel, this ratio is equal to 
a i 


that of AG to AD, or of AK to AA’, a constant ratio, since 
these last two lengths are constant. 

The directrix DE corresponds to the focus F' and the direc- 
trix D'E' to the focus F". 


266. TurorEM II. — The section of a right circular cone by a 
plane is a curve of the second degree. 


Draw through the axis of the cone a plane perpendicular to 
the secant plane; this plane intersects the cone in the two gen- 
eratrices SG, SH, and the secant plane in the straight line 
AA’. 

1° Consider now the case when the straight line Ad’ inter- 
sects the two generatrices SG and SH, on the same side of the 
vertex S (Fig. 161). 

Describe two circles O and O' which are tangent to the 
straight line AA! and to the two elements SG', SH". If the 
figure be revolved about the ; 
axis SO’, so that the element 
SG' will generate the cone, 
the two circles will generate 
two spheres, which are tan- 
gent to the cone along the 
circles of contact. GH, G'H'. 
The secant plane is tangent 
to one of the spheres at the 
point F, since it 1s perpen- 
dicular to the radius vector 
OF at its extremity; it is 
also tangent to the other 
sphere at the point F”. 

Let M be any point of the Fig. 161. 
section made by the secant plane; the generatrix SM which 
passes through this point is tangent to the spheres at the 





282 PLANE GEOMETRY. BOOK III. 


points Z and L'; draw the straight lines MF and MF". The 
straight lines MF and MZ are equal, since they are tangents . 
drawn from the same point M to the sphere O; the straight 
lines MF" and ML' are equal, since they are tangents drawn 
from the point M to the sphere O'; one has, therefore, 


MF + MF'= ML + ML'= LL". 


But the portion DL! of the generatrix comprised between the 
parallel circles GH, G'H' is constant and equal to GG'; there- 
fore the sum of the distances of each of the points of the curve 
from the two fixed points # and F" is constant, and, conse- 
quently, this curve is an ellipse whose foci are F' and F". 

The constant sum GG' is equal to the major axis AA’. If 
A'K be drawn through the, point A’ parallel to GH, one de- 
termines on the generatrix a length Adv equal to the focal 
distance FF’; for if from the equal lengths GG, AA’ one sub- 
tracts on the one hand the equal lengths AG and AG’, on the 
other the equal lengths AF and A'F"', it follows that the 
lengths AW and FF" are equal. 

Let us consider the straight lines DE and D'E', the intersec- 
tions of the secant plane with the planes of contact GH, G'H". 
If from the point M a perpendicular 
MP be dropped on the major axis, the 
distance of the point M from the straight 
line DE is equal to PD. Let NMN' 
be the parallel circle whieh passes 
through the point MW; the length MF 
or ML is equal to GN on account of 
the parallels DG, PN; one has 


ON AG ea 

DP AD AA’ 
Whence the distance of each of the 
points of the ellipse from the focus / 
and from the straight line DE are to 
each other as the distance between the 
foci to the major axis. This straight line DE is a directrix 
of the ellipse; the straight line D'Z' is the second directrix. 











Fig. 162. 


CHAP. VIII. THE CONIC SECTIONS. 283 


92° When the straight line AA’ intersects the two gener- 
atrices SG and SH on opposite sides of the vertex S (Fig. 162), 
one has 

MF' — MF = ML! — ML=LL'= GG". 

The difference of the distances of each of the points of the 
curve from the two points F and F" is constant; this curve is 
a hyperbola whose foci are the points # and F’. The straight 
lines which are the intersections of the secant plane with the 
two planes of contact are the directrices of the hyperbola. 

3° Finally, suppose that the straight line AA! is parallel to 
the element SH (Fig. 163). Construct a sphere tangent to the 
cone along the circle GH and to the secant plane at F. Let 
DE be the intersection of the secant 
plane and the plane of the circle of 
contact. Through the point M of 
the section draw the straight line 
ME perpendicular to DE, and the 
generatrix SM, which intersects the 
curve of contact in L; the straight 
line ME will be parallel to 4A’ and 
to SH; therefore the three straight 
lines ME, SM, SH lie in the same 
plane, and the three points H, L, # 
are on the straight line which is 
the intersection of the plane of contact with the plane just 
mentioned. The two triangles MLE, SHL are similar; since 
SL is equal to SH, one has also MZ equal to ME; but ML 
equals to MF, because they are tangents drawn from the. point 
M to the sphere; consequently MF is equal to ME. There- 
fore the curve is a parabola of which the point #’ is the focus 
and DE the directrix. 

This elegant method for finding the properties of the foci 
and the directrices of the curves of the second degree, is due to 
DANDELIN. 








Fig. 163, 


267. To place a curve of the second degree on a given cone. 


1° The curve is an ellipse. In the triangle AA'K (Fig. 
161), one knows the two sides AA’, AK, which are the major 


284 PLANE GEOMETRY. BOOK IIL. 


axis and the distance between the foci, and also the angle 
opposite AA', which is the complement of half the angle at the 
vertex of the cone. Since the major axis is greater than the 
focal distance, this triangle can always be constructed; the per- 
pendicular at the mid-point of A'A determines the point S, 
and consequently all that determines the position of the secant 
plane. 

2° The curve is a hyperbola. In the triangle AA'K (Fig. 
162), one knows likewise two sides, and also an angle opposite 
one of them, but since the side opposite the given angle is the 
shortest, the construction of a triangle is not always possible. 
It is necessary that one has a >c cos y (2 a being the transverse 
axis, 2c the distance between the foci of the hyperbola, 2 y the 


angle at the vertex of the cone); whence cos y <= and conse- 


quently cos y << cos 6, when @ is the angle between the major 
axis and the asymptote; therefore the angle between the 
asymptotes should be less than the angle of the cone. 

3° The given curve is a parabola. On joining the center O 
of a sphere to the point G, a right triangle OAG (Fig. 163), in 
which one knows the side AG which is the semi-parameter 
of the parabola, and the angle OAG the complement of 6. 
Having constructed this triangle, one constructs OS perpen- 
dicular to OA, and produces it until it intersects AG; the 
moment the distance SA is known, the problem is solved. 

To sum up, one can place on a given cone every ellipse, every 
parabola, and every hyperbola in which the angle between the 
asymptotes is less than the angle of the cone. 


268. Remark. — Suppose that the spheres used in the pre- 
ceding discussion be always inscribed in the cone so that they 
intersect the secant plane; for this it is sufficient that the 
generating circles be tangent to SA, SA', and intersect AA’; 
the intersections of the spheres by the secant plane are circles, 
and one knows that, in case of. the ellipse or of the hyperbola, 
the sum or the difference of the tangents drawn to these circles 
from any point of the curve is constant; that, in case of the 
parabola, the tangent drawn to the circle from any point of 


CHAP. -VIit. . THE CONIC SECTIONS. 285 


the curve is equal to the distance of this point from a certain 
straight line. 

The Greek geometers knew the curves of the second degree 
as sections of a cone with circular base by a plane. APouLuo- 
nius (247 B.c.) wrote a treatise of eight books on conic sections, 
in which he gave an account of what had been discovered 
before his time, and gave an exposition of his discoveries con- 
cerning this subject. The treatise of APoLLontus contains 
the fundamental properties of conic sections; we may men- 
tion especially in this connection the two theorems concerning 
conjugate diameters (§§ 162, 163, and 191), the properties 
concerning the asymptotes of the hyperbola, the elementary 
properties of the foci. 


286 PLANE GEOMETRY. BOOK III. 


CHAPTER IX* 
THE DETERMINATION OF THE CONIC SECTIONS. 


269. The general equation of the second degree 
Aa’ + 2 Bey + Cy? +2 De+2 Ey+ F=0 


contains six coefficients; but since all of the terms may be 
divided by one of the coefficients, provided that this coefficient 
be different from zero, the equation will involve but five arbi- 
trary parameters, which are the ratios of five coefficients to 
the sixth. In order to determine a curve of the second degree, 
it is necessary to assign values to the five parameters, or, better, 
that the five parameters should satisfy five relations; but in 
this case it is necessary to examine whether the five equations 
of condition have a system of real solutions, and if, moreover, 
the corresponding equation of the second degree represents a 
curve. If the five equations of condition have a system of 
real solutions possessing this property, there will be a curve of 
the second degree satisfying the proposed conditions. 

In general, the relations between the parameters correspond 
to geometric conditions which the curve must satisfy. Thus, 
one can require the curve to pass through given points, to 
be tangent to given straight lines, etc. One will express the 
condition that the curve pass through a given point on requir- 
ing that the co-ordinates of the point satisfy the equation of 
the curve, which leads to a relation of the first degree between 
the coefficients. The condition that the curve is tangent to a 
given straight line will be found by requiring that the equation 
which determines the abscissas of the points of intersection of 
the curve and the straight line has two equal roots, which 
gives a relation of the second degree between the coefficients. 
A geometric condition which must be expressed by two rela- 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 287 


tions will be regarded as double. If, for example, one should 
require the curve to touch a given straight line at a given 
point, the equation which will furnish the abscissas of the 
point of intersection and of the curve, should have two roots 
equal to a given quantity; whence there will result two rela- 
tions of the first degree between the coefficients; the geometric 
condition stated ought, therefore, to be reckoned as two single 
conditions. Accordingly, it is necessary to have five geometric 
conditions in order to determine a curve of the second degree. 

If one know that the curve is a parabola, the coefficients 
must satisfy the relation AC — B’=0; the equation will con- 
tain but four arbitrary parameters and the parabola will be 
defined by four conditions. 

Similarly, if one know that the curve is an equilateral hyper- 
bola, it will be necessary that the two straight lines repre- 
sented by the equation Aa’ + 2 Bry + Oy? = 0, straight lines 
parallel to the asymptotes (§ 150), be perpendicular to each 
other, which gives a relation between the coefficients; when the 
axes of co-ordinates are rectangular this relation is 4 + C= 0. 
Four conditions are sufficient, therefore, to determine an equi- 
lateral hyperbola. 

Before proceeding farther, it is best to generalize the defini- 
tions, in order to avoid the restrictions, which would introduce 
imaginary solutions in the statement of theorems. 


Pornts AND IMAGINARY STRAIGHT LINES. 


270. A system of real values of x and y determine a point 
in a plane; in an analogous manner we call an imaginary 
point a system of imaginary values assigned to zandtoy. If 
two systems of imaginary values be of the form x=a-+ bi, 
y=e+di, and s=a—bi, y=ec—di, we say that the two 
imaginary points are conjugate. 

An equation of the first degree, Aw + By + C = 0, with real 
coefficients, is satisfied by the co-ordinates of an infinite num- 
ber of real points, whose locus is a straight line; but it is 
satisfied also by an infinity of systems of imaginary values 
assigned to « and y; because if any imaginary value be 


288 PLANE GEOMETRY. BOOK III. 


assigned to a, the corresponding value deduced for y will be 
imaginary ; if two conjugate imaginary values be given to a, 
the two corresponding values of y will also be conjugate. 

In an analogous manner, we call an imaginary straight line 
the ensemble of solutions of an equation of the first degree 
with imaginary coefficients. It is to be noticed that an imagi- 
nary straight line passes through one real point. Let, in fact, 


(A! +A") 2 + (B' + Bl) y + (C'+ OC") =0, 
(A'x + Bly + C) +i (Ale + Bly + C") =0, 


be an imaginary straight line. This equation is satisfied by 
the co-ordinates of the point of intersection of the two real 
straight lines 


Ae + By + C!=0, Ave + Bly + 0" =0. 


or 


In the case of the general equation of the first degree, involv- 
ing three coefficients and consequently two arbitrary parameters, 
two points, real or imaginary, will determine the straight line. 
If a’, y', wv", y", be the co-ordinates of two given points, the 
straight line which passes through these points will have as its 
equation 

eo —a yy —y’ 
gl! — xy! y"' ee y! 





The straight line which passes through two conjugate 
imaginary points is real. Let, in fact, 7’ = a + bi, y'= ¢ + di, 
a'—a— bi, y'=c—di; the equation of the straight line 


reduces to 
e—-a y—e 
b d 





The point which has the co-ordinates 
a! ae gl! y! + yl" 
eee 

will be called the mid-point of the straight line which joins 


the two given points; in case the two points are conjugate 
imaginaries, the mid-point is a real point. 





CHAP. IX. DETERMINATION OF CONIC SECTIONS. 289 


An algebraic equation f (2, 7) = 0, with real coefficients, is in 
general satisfied by the co-ordinates of an infinitude of real 
points constituting a curve; it is also satisfied by the co- 
ordinates of an infinitude of imaginary points, conjugate in 
pairs. If the coefficients be imaginary, the equation has 
always an infinitude of imaginary solutions, but only a limited 
number of real solutions; the totality of these solutions will 
form what we call an imaginary curve. 

Two equations, the one of the first degree, the other of the 
second degree in # and y, have two solutions. It is said, there- 
fore, that a straight line intersects a curve of the second 
degree in two points, real or imaginary. A real straight line 
intersects a real curve of the second degree in two points, 
which are real or conjugate imaginaries. This suffices to 
explain a fact which has already presented itself several 
times; when one seeks, for example, the locus of the mid- 
points of a series of parallel chords in an ellipse, one finds by 
calculation an indefinite straight line, and yet the locus which 
is defined geometrically is composed only of that position which 
lies within the ellipse; the external secants intersect the ellipse 
in two conjugate imaginary points; the mid-point of the chord 
is, moreover, a real point, and the diameter is thus prolonged 
without the curve. 


CONCERNING THE INTERSECTION OF TWO CURVES OF 
THE SECOND DEGREE. 


271. We remark, first, that if the two curves coincide, that 
is, if the two equations are satisfied by the same systems of 
the variables x and y, the coefficients are proportional. In 
fact, the equations 
(1) Cy? + 2(Bu + E)y + (Aa? + 2 Da + FP) =9, 

(2) C'y? ++ 2(Bla+ E\y+ (Ai? +2 D+F) =O, 
have the same roots for the same value of #; one has 


_ Be+E _ Ag? +2 De+ FF 
~ Bete Al’? +2Da4+F" 





¢ 
C! 


290 PLANE GEOMETRY. BOOK III. 


and, since this relation should exist whatever x may be, one 
deduces 

Cet oe 

Ch ek Ae 
The converse is true; in case the coefficients are proportional, 
the two equations will be identical and the two curves coincide. 

We suppose, in what follows, that the curves are different, 

that is, that the coefficients are not proportional. We consider 
first the case when the two coefficients C and C’ are different 
from zero; if the equations be subtracted member from mem- 
ber, after having been multiplied respectively by C' and C, one 
eliminates y? and obtains an equation of the form 


(3) 2(Bywe+ LE\)y+ (Aja? + 2 Dw+ F))= 0, 


which, with equation (1), forms a system equivalent to the 
system of the two given equations (1) and (2). The five coef- 
ficients B,, E,, A,, D,, F,, cannot all be zero at the same time; 
because if that were the case the coefficients of the two 
equations (1) and (2) would be proportional. If the two 
coefficients B, and E, were zero, equation (3) would become 
Aw’+2D e+ F,=0; it would furnish two values for 2; to 
each of which, by reason of equation (1), would correspond 
two values of ¥; in all, four solutions. Suppose that the two 
coefficients B, and EL, were not zero at the same time; in gen- 
eral, the value n= 7) vhich annuls the coefficient of y in 
equation (1), does not reduce the polynomial 4,2’ +2 Dw + F, 
to zero; in this case, the quantity Bw + E, being different 
from zero for all of the solutions of equation (3), this equation 
can be put under the form 


Ayw+2Da+F,. 
“) 1 Bee 
on substituting this value of y in equation (1), an equation of 
the fourth degree is found, 
(5) apt + aa? + age”? + ast + ay = 0, 


which, combined with equation (4), forms a system equivalent 
to the system of two equations (1) and (3), and, consequently, 





CHAP. IX. DETERMINATION OF CONIC SECTIONS. 291 


to the proposed system. The five coefficients of equation (5) 
cannot all be zero at the same time, because, if that were the 
case, equation (5) becoming an identity the two proposed equa- 
tions would be satisfied by all the systems of values of # and y 
which would satisfy equation (4) or equation (3); the two 
curves would coincide with the curve represented by equa- 
tion (3) and, consequently, would have proportional coeffi- 
cients. Equation (5) gives four values for 2; to each value of 
which there corresponds, owing to equation (4), one value of y, 
which furnishes four solutions of the given system. 


If the value « = — = annul the polynomial 4,2? + 2 Dw +F,, 
equation (3) can be put under the form 
(By + E))(y + ma + n) = 0, 
and decomposes into two distinct equations, Byw+ E,=0, 


y+me+n=0; the first gives the value x = — a to which, 
a 
owing to equation (1), correspond two values of y; from the 


second, one gets y= — ma —n; and, by replacing y by this 
value in equation (1), one obtains an equation of the second 
degree in 2, which furnishes two new solutions; in all, four 
solutions. Moreover, it can happen that this last equation of 
the second degree in w reduces to an identity; in this case, the 
co-ordinates of every point of the straight line y + mz +n=0 
satisfy the two proposed equations, which represent pairs 
of straight lines, two of which coincide. 

In case one only of the coefficients C and C' is zero, one of 
the proposed equations will be of the form (3), and the dis- 
cussion just given will be repeated. 

Let us consider now the case when the two coefficients C 
E' 
RB’ 
which annuls the coefficient of y in equation (2), does not 
reduce the polynomial <A's?+ 2 D'x+ F' to zero, one finds 
from this equation _ 


Y= 


and C’ are zero at the same time. If the value x= — 


_ Ale? + 2 Die + F! 
2(Be+E)”’ 





292 PLANE GEOMETRY. BOOK IIl. 


and, by substituting in equation (1), an equation of the third 


degree in x will be obtained, which gives three solutions. Ifthe 


value «= — *. reduces the polynomial A’x’ + 2 D'x + F’ to 


zero, equation (2) may be written 
(Bie + E'\(y + mx +n) =0, 


and represents two straight lines B'x + E'=0, y+ mz+n=0, 
the first of which intersects the curve (1) in one point, the 
second in two points. It happens that one of these straight 
lines belongs to curve (1), and in this case the proposed equa- 
tions represent pairs of straight lines, two of which coincide. 

From what precedes, it follows that two curves of the second 
degree cannot have more than four points in common, at least 
that these curves consist of pairs of straight lines, two of 
which coincide. In case the two given equations have real 
‘coefficients, their points of intersection are real or conjugate 
imaginaries. 


272. It is easy to form equation (5), which, in the general 
case, determines the abscissas of the four points of intersection 
of two curves of the second degree. Let Ayy’ + Ay + A, = 9, 

iy? + Aly + A', = 0, represent the two given equations, in 
which A, and A‘, designate constants, A, and A’) polynomials 
of the first degree in a, A, and A’, polynomials of the second 
degree in « On subtracting these equations, meinber from 
member, having multiplied them respectively by A’) and Ap, 


one has 
(A, A‘, — A'A;) y + A, A', — AA, = 0. 


On'multiplying by A, and A,', subtracting, and suppressing the 
factor y, one has, in a similar manner, 


(A, A's aE A',A;) y + (A,A a — A! As) = (0. 


The elimination of y between the last two equations leads to 
the equation of the fourth degree 


(A,A F foe A A) (A, A’, a A!',A,) — (A,A f srs A',A,)? = 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 293 


273. CoroLttAry. — An equation of the second degree, with 
imaginary coefficients, cannot have more than four real solu- 
tions. In fact, the first member of the equation has the form 
S +iS,, S and S, representing real polynomials of the second 
degree; if the equation is satisfied by real values assigned to x 
and to y, one will have separately S=0, S,=0; the real points 
of the locus are therefore the points of intersection of the two 
real curves S = 0, S,=0, and these points are in general four 
-in number. 

There is an exception, namely, the case cited above, when 
the curves consist of pairs of straight lines, two of which coin- 
cide; in this case the equation of the second degree represents 
two straight lines, one of which is real. Finally, if the two 
curves S = 0, S,;=0 coincide, the first member of the given 
equation will be divisible by a constant imaginary factor and 
the coefficients of the equation become real. 


274. Lemma. — Every system of n homogeneous equations of 
the first degree involving n +1 unknown quantities, is satisfied by 
an infinitude of systems of values of the unknown Wadlslalaes: one 
of which at least is different from zero. 


Let us consider first an equation involving two unknown 


quantities, 
ax + by = 0. 


If the two coefficients a and b were zero, the equation would be 
satisfied by arbitrary values of « and y. Suppose that one of 
the coefficients, for example b, be not zero; the equation can be 


put under the form y= ~<a, and an arbitrary value be as- 


signed to 2; to each value of « corresponds one value of y. 
Thus the given equation is satisfied by an infinity of systems 
of values of « and y, one of which at least is different from 
zero. 

Consider now two equations involving three unknown quan- 


tities, 
ax + by + cz =), 


ale + bly + cz =0 


294 PLANE GEOMETRY. BOOK III. 


If the six coefficients be all zero at the same time, the equa- 
tions would be satisfied by all possible arbitrary values of a, 
y, 2. Suppose that one at least of the coefficients, for example 
c, be not zero; the system of the two given equations could be 
replaced by the equivalent system 


eS __ ax + by 
ee 
(ac! — ca') a + (bc! — cb')y = 0. 


In accordance with what we have said, the second equation is 
satisfied by an infinity of systems of values of a and y, one of 
which at least is different from zero; to each of them corre- 
sponds one value of z given by the first equation. Thus the 
system of the two given equations is satistied by an infinity of 
systems of a, y, z, one of which at least is different from zero. 

The same reasoning can be continued indefinitely. Suppose 
that we have the following three equations involving four 
unknown quantities : 


ax +by + cz 4+ dt as 
ale +b'y +c%+d't =0, 
ale t+ b"y+ clz+d"t=0. 


If the twelve coefficients were all zero at the same time, the 
equations would be satisfied by all possible arbitrary values of 
x, y, 2, t. Suppose that one of the coefficients at’ least, for 
example d, be not zero; the system of given equations could be 
replaced by the equivalent system 

ax + by + cz 
ae : ; 





een 


(ad' —a'd)x + (bd' — b'd)y + (cd' —c'd)z = 0, 

(ad" — ad) x + (bd" — b"d)y + (cd"" — c''d)z = 0. 
By reason of the preceding discussion, the system of the last 
two equations is satisfied by an infinitude of values of a, y, z, 


one of which at least is different from zero; to each of them 
corresponds one value of ¢ given by the first equation. 


“ 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 298— 


275. TuEeorEM I.— Through five given points, no three of 
which lie on a straight line, a curve of the second degree can be 
passed, and one only. 


Call (a, Y1)5 (®2y Yo)> (®3y Ys)s (®ay Ya)» (Ws Ys) the co-ordinates 
of the five given points. In order that the curve of the second 
degree 


(1) Aa? + 2 Bay + Cy? +2 Dx+2Hy+ F=0 


pass through these five given points, it is necessary and 
sufficient that the five equations 


( Ag? + 2 Bay, + Cy? + 2Dx,+2Ey+ F=0, 
Aa? + 2 Bays + Cy? +2 Da, +2 Ey,+ F=0, 
(2) 1 Ax? + 2 Basy; + Cy? + 2 Dry +2 Ly, + F = 9, 
Axe + 2 Bey, + Cy? +2 Dau +2 Hy, + F=0, 
| Aas? + 2 Basys + Cy; + 2 Da; +2 Ey; + F= 0, 





be satisfied. We have thus five equations, homogeneous and 
of the first degree, between the six unknown quantities A, B, 
C, D, E, F. It follows from the preceding lemma that these 
equations are satisfied by an infinitude of systems of values 
of the unknown quantities A, B, C, D, E, F, one of which at 
least is different from zero. We notice that in none of these 
solutions, the first five coefficients are all zero at the same time, 
because then, by virtue of any one of equations (2), one would 
have F=0. We remark further that if the five given points 
be not on a straight line, the first three coefficients A, B, C 
cannot be zero at the same time; because equation (1) would 
be reduced to the first degree, and would represent a straight 
line passing through the five points. On assigning to the six 
coefficients the values which constitute one of the preceding 
solutions, one obtains a curve of the second degree passing 
through the five given points. Thus, through five given 
points one curve of the second degree at least can be made to 
pass. 

It follows that if of the five given points no three lie on 
a straight line, one can pass through these five points one, 
and only one, curve of the second degree; because if one could 


296 PLANE GEOMETRY. BOOK III. 


pass two, these two curves would have five points in common; 
but we have seen (§ 271) that two curves of the second degree, 
which are not composed of straight lines, cannot have more 
than four points in common. 

The different solutions of the system of equations (2), in 
order to determine the same curve of the second degree, are 
formed from proportional quantities. Having learned that 
one of the undetermined coefficients is different from zero, one 
seeks ‘the ratios of the other five to it, and will need to solve 
a system of five equations of the first degree in five unknown 
quantities. 

The same conclusions are applicable to the case where three 
given points, and only three, are on a straight line. The locus 
of the second degree is composed of this straight line and the 
one which passes through the other two points. 

If four of the points be on a straight line, the problem is 
indeterminate. The locus of the second degree is composed of 
this straight line, and of any straight line passing through the 
fifth point. 


276. KemMARK.— One can, by aid of a determinant, form the 
equation of the second degree which passes through five given 
points. Consider, for this purpose, the determinant 


wy 6Y 
1, HY Yi % 
U, Life Yo ea Yo 
es Ys Ys % Ys 
U4 Ue Yu Ue Ys 
5 UYs Ys Ws Ys 


ee 








This is an integral polynomial of the second degree with 
respect to the variables 2 and y. It becomes zero in case 
xz and y are replaced by 2, and y,; because then the elements 
of the first horizontal line become equal to those of the second. 
The same is true if x and y be replaced by 2 and y, and so on. 
Whence it follows that the equation A= 0 represents a curve 
of the second degree passing through the five given points. 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 297 


277. Corottary I.—A quadrilateral abcd (Fig. 164) being 
given, represent the equations of the two opposite sides 
ab, cd by «=0, B=0, those of the 
other two opposite sides be, ad by 
y=0, 8=0; the equation 


(3) Auf + By’ = 0, 


in which the coefficients a and 6b are 
arbitrary, represents all the curves of 
the second degree which pass through 
the four points a, b, c,d. The letters Fig. 164. 

a, B, y, 8 representing polynomials of 

the first degree in w and y, the equation is of the second degree ; 
the co-ordinates of the point a, the intersection of the straight 
lines ab and ad, reduce the two polynomials @ and 6 to zero, 
and consequently the first member of equation (3); the same 
is true of the other three points b, c, d. Whence whatever 
the value of the coefficients A and B may be, the curve 
represented by equation (3) passes through the four points 
a,b,c, d. This equation represents every curve of the second 
degree which passes through the four points, because a fifth 
point e determines the curve and one can assign to the ratio 
of the coefficients such a value that the curve will pass through 
this fifth point taken at random in the plane. 

_ Equation (3) has a very simple geometrical signification: 
the polynomials «, 8, y, 5 being proportional to the distances 
of any point (#, y) from the sides of the quadrilateral, it fol- 
lows that the product of the distances of any point of the curve 
from the two opposite sides ab, cd of the inscribed quadrilateral 
is to the product of the distances of the same point from the two 
opposite sides be, ad in a constant ratio. The value of this 
ratio determines the curve. 

In general, if the equations of two curves of the second 
degree be represented by S=0, S,=0, the equation S +kS,=0, 
in which & is an arbitrary parameter, represents every curve 
of the second degree which passes through the four points 
common to the first two. 





298 PLANE GEOMETRY. BOOK Itt. 


278. CoroLtLARy II.— We propose to determine a parabola 
which passes through four given real points a, 0, ¢, d. If these 
points be connected two and two by two straight lines ad, cd, 
which intersect and which are chosen as axes of co-ordinates, 
the general equation of the second degree which passes through 
these four points is 


(4) (E+¥—1) 548-1) — key = 0; 


a and b being the abscissas of the points a and 0, c and d the 
ordinates of the points c and d. In order that the locus be a 
parabola, it is necessary that the parameter k satisfy the con- 
dition : 
~ ee oe 
(5) (i =, i) Caan 

In case the product abcd is negative, one finds two imaginary 
values for k, and it is impossible to pass a real parabola 
through these four points. If the product be positive, one 
concludes that a convex quadrilateral having the four points 
as vertices can be formed, and obtains two real and different 
values for k, and, consequently, two real curves of the genus 
parabola passing through the four points. In case the points 


could be connected two and two by parallel straight lines, each 
pair of parallel straight lines would constitute a solution. 


279. CorotuAry IIT. —It is easy to form the general equa- 
tion of curves of the second degree, which pass through three 
given points a, b, c. If a=0, B=0, y=0, represent the 
equations of the three straight lines bc, ca, ab, the equation 


(6) ABy + Bya + Cas = 0 


represents a curve of the second degree passing through the 
three given points. This equation involves two arbitrary 
parameters, the ratios of two of the coefficients to a third, and 
one could so dispose of the two parameters as to make the 
curve pass through two additional points chosen at random 
in the plane. 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 299 


280. TuHrorem II.— One curve of the second degree can be 
drawn tangent to two given straight lines, at two given points, and 
made to pass through another given point, and only one. 


In order that a curve of the second degree 
(7) f(@, y) = Av’ + 2 Bay + Cy +2 De+2Ey+Fr=0 
be tangent to a straight line 


(8) a (2% — 2%) + by — H)= 9, 


at a point (2,, y,), it is necessary and sufficient that the curve 
pass through the point (a, y,), and that the angular coeffi- 
cient of the tangent at this point be equal to that of the straight 
line, which furnishes two equations, 


(9) S(t, w= 9, Of a (1, 1) — af, (2, 71) = 0, 


which are homogeneous and of the first degree in the coeffi- 
cients A, B, C, D, H, F. 

We have thus five equations which are homogeneous and of 
the first degree. We have learned that such a system of equa- 
tions has an infinitude of solutions in which one at least of the 
coefficients, say F, is different from zero; to one of these solu- 
tions there corresponds a curve of the et degree satisfying 
the required conditions. 

There cannot exist more than one curve of the second degree 
satisfying these conditions, because, if there were two, the 
equation of the fourth degree which one obtains when one 
seeks their common points would have two double roots and 
one single root, which is impossible. 


281. Corotitary I.— The equation 
(10) a3 —ky =0, 


in which k is an arbitrary parameter, represents every curve of 
the second degree tangent to two lines a=0, B=0, at the 
points where they are intersected by the straight line y= 0 
(Fig. 165). The curve is tangent to the first straight line; 


300 PLANE GEOMETRY. BOOK III. 









for if one makes «=0 in the equation of the curve, one has 
y’ = 0, and consequently the two points of intersection of 
the straight line and the curve coin- 
cide. The curve is in a similar man- 
ner tangent to the second straight 
| line, and the two points of contact 
are situated on the straight line 
oe. y=0. Equation (10) represents every 
curve possessing these properties; 
Dia ts a because the parameter k can be so 
sees determined that the curve can pass 
through any other point chosen at random in the plane. 

This equation signifies that the product of the distances of any 
point of the curve from two tangents is to the square of the dis- 
tance of this point to the chord of contact in a constant ratio. 

In general, if the equation of a curve of the second degree be 
represented by S=0, the equation S — ky? = 0 will. represent 
every curve of the second degree tangent to the first at two 
points situated on the straight line y = 0. 


CorotiAry II. — One can determine the parameter k by the 
condition that the curve be a parabola. If the two tangents be 
chosen as axis of co-ordinates, equation (10) becomes 


2 
(11) (Z+$-1) -2hay =0. 


ft 
v 


In order that the curve be a parabola, it is necessary that the 
condition 


be satisfied; which furnishes the two solutions k=0,k =4; 


the straight line ad corresponds to the first; a parabola, whose 
equation can be put under the form 


x ie te 
Et 1=0, 


corresponds to the second. 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 301 


282. AppiicaTion.— As an application, form the equation 
of the second degree which represents the ensemble of the two 
tangents drawn from a point p with the co-ordinates x, % 
(Fig. 165) to a conic whose equation is 


f(a, y= Aol +2 Bay + Cy +2Dx +2 Ey + F=0. 


The equation of the chord of contact ab is, as we have seen 


(§ 125), 
(a) y=(Aaq,4+ By,+D) e+ (Ba,+ Cy, +E) y+ Dx,4+ £y,+ P=), 


the first member of which is designated by y. The two tan- 
- gents pa and pb represent a conic doubly tangent to the given 
conic f(a, y) =0 at the points situated on the straight line; 
they are therefore represented by an equation of the form 


(0) Sf (a, y)— ky’ = 9, 


where & represents a constant coefficient which remains to be 
determined. For this purpose it is sufficient to express the 
condition that the curve (b) passes through a point taken on 
the conic formed by the two tangents pa and pb; we express 
the condition that it passes through the point p whose co-ordi- 
nates are (2, 7), that is, that equation (b) is satished by # = a, 
y=y, If in y one put e=x, y= yy, y reduces to f (21, J); One 
has, therefore, the condition 


S(@y 1) —kf? @, w= 0, 





which gives for k the value , and the equation re- 


S (@y 1) 
quired is 
(c) Sf (@, y) f @y %)— y' = 9, 


where y should be replaced by the expression (@). This equa- 
tion is called the quadratic equation of the tangents drawn from 
the point X, Y. 


282. 2. We confine ourselves to stating the following re- 
sults, which it is easy to verify. (See §§ 303 and 331.) 


302 PLANE GEOMETRY. BOOK III. 


Let «= 0, B= 90, y = 0 be the equations of the three sides 
of a triangle. The general equation of the conics inscribed in 
this triangle is 


Nae? + wB? + vy — 2 pvBy — 2 Avye — ZApaB =0, 


A, #, v representing the variable parameterg, This equation 
can be written in the irrational form 


(Vr + V a8 + Vry)(-Vra —V pB +V vy) 
(Vra +VpB —Vvy)(Vira —VpB —V vy) = 


The general equation of the conic inscribed in the aes 
lateral whose sides have the equations 


at pt+y=9, a—B+y=09, a+B—y=0, e—B—y=0), 


2 2 
is Bal B —7y=)0, 





d being a variable parameter. (See § 331, Examples.) 


MULTIPLE CONDITIONS. 


283. Let us examine the geometric conditions by which a 
curve of the second degree may be defined. Thus far we have 
mentioned none other than single conditions, such as points 
and tangents. The center is equivalent to two conditions; 
because if the center be taken for the origin of co-ordinates, the 
equation of the second degree, being deprived of the terms of 
the first degree, cannot contain more than three arbitrary 
parameters; thus the curve is defined by its center and three 
points. 

A diameter, with the direction of the chords, is equivalent 
to two conditions; because if the diameter be taken for the 
x-axis and a line parallel to the corresponding chords for the 
y-axis, the equation, being deprived of the two terms of 
the first degree in y, does not contain more than three arbi- 
trary parameters. 

A system of conjugate diameters is equivalent to three con- 
ditions; because if they be taken for the axes of co-ordinates, 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 3803 


the equation being reduced to the form aa’ + by? + ¢ = 0, con- 
tains only two arbitrary parameters. In general, let «a= 0, 
B=0 be the equation of two conjugate diameters; the dis- 
tances « and B of each point from the two conjugate diameters 
being proportional to the co-ordinates of this point with respect 
to these diameters, the curve will be represented by the 
equation 

(12) aw + bp? +¢=0)0, 


with two arbitrary parameters. 
The equation 
(13) e+ kpB=0 


is the general equation of the parabolas of which the straight 
line « = 0 is a diameter, and the straight line B = 0 the tan-— 
gent at the extremity of this diameter. 

One knows that the equation of the hyperbola, referred to 
its asymptotes as axes, is xy =k. In general, leta=—0, B=0 
be the equations of two asymptotes; the hyperbola will be 
represented by an equation of the form 


(14) ap —k = 0, 


which contains but one arbitrary parameter k. Thus the two 
asymptotes are equivalent to four conditions, and the curve is 
determined by two asymptotes and a point or a tangent. If 
one were given but one asymptote whose equation is a = 0, the 
equation 8 =0 of the other asymptote being indeterminate, 
equation (14) would contain three arbitrary parameters, so that 
one asymptote is equivalent to two conditions. 

We have seen that every equation of the second degree has 
one focus and one directrix; whence it follows that the 
equation 


(15) (a — a)’ + (y — f)* "(ma + ny +h)’ = 0, 


by which the focal property is expressed in rectangular co- 
ordinates, and which involves five arbitrary parameters «, £, 
m, n, h, represents every curve of the second degree. A focus 
is equivalent to two conditions; because, if one were given a 
focus, its co-ordinates « and 8 being known, equation (15) 


304 PLANE GEOMETRY. BOOK III. 


would contain but three arbitrary parameters. Similarly, one 
directrix is equivalent to two conditions; because, on being 
given the equation of the directrix, the ratio of the three 
parameters m, n, h to a third is determined. 

The results that we have obtained may be derived in another 
manner. It is clear that the two co-ordinates of a special 
point of a curve of the second degree, like the center, a focus, 
a vertex, etc., are determined when the coefficients of the equa- 
tion of the second degree are known, and consequently that 
there exist two equations between these co-ordinates and the 
coefficients; if therefore such a point be given, one will have 
two relations between the coefficients. A similar discussion 
applies to the two parameters of a special straight line, such 
as the directrix or axis, etc.; if this straight line be given, one 
will have, moreover, two relations between the coefficients. 

Thus, for example, if f(a, y)=0 be the equation of the 
curve, one can express the condition that a given point is the 
center by requiring that its co-ordinates satisfy the two equa- 
tions f', =0, f',=90. In order to express the condition that a 
given point is a vertex, it is sufficient to require that its co- 
ordinates satisfy the equation of the curve and that the normal 
at this point passes through the center. 

It is to be noticed that the preceding forms under which the 
equation of the second degree have been put, reduce to the 
form «8 — ky? = 0, composed of three polynomials of the first 
degree a, 8, y, of which the first two represent tangents drawn 
from an arbitrary point p of the plane, and the third represents 
the chord of contact. If the point p coincide with the center 
of the hyperbola, the tangents « and 8 are the asymptotes ; if 
the chord of contact be removed to infinity, the polynomial y 
reduces to a constant, and the equation #8 — ky’ = 0 becomes 
03 —k=0. Equation (12), put under the form 


(aVa + BV —b) (aVa— BV —b) +¢=0, 


reduces to equation (14). 


284. Tur DreTERMINATION OF THE Focr oF THE CONIC. — 
Let a, B be the co-ordinates of a focus of a conic whose equa- 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 305 


tion is f(a, y). We have seen (§ 216) that the equation of the 
curve in rectangular co-ordinates can be written in the form 


(15) (x — a)? + (y — B)’ — (ma + ny +h)’ =0. 
This equation can be written 


[yy —B)—i@—«)]Ly—B)+i@—a)]—(mat+ny +h)’ =0, 


which is of the form 
PQ— R’=0, 


P, Q, R representing three linear functions in w and y. It 
follows, therefore, that the conic represented by equation (15) 
is tangent to the two straight lines 


(a) y—Bt+i(a—a)=0, y—B—i(a—«)=0, 
the chord of contact being the directrix 
mx +nyth=0. 


Two straight lines (a) pass through the focus (@, 8) and have 
the angular coefficients + 7 and —7i; since they are tangent to 
the curve, they are the two tangents drawn from the focus 
(a, 8) to the conic. It follows, therefore, that the focus is a 
point such that the two tangents drawn from this point to the 
conic have the angular coefficients + 1; the directrix is the chord 
of contact. : 

One can say also that the ensemble of the two tangents 
drawn from the focus (a, 8) to the conic has the equation, in 
rectangular co-ordinates, 


(«—a)’?+ (y— Bp)? =), 


an equation identical with that of a circle of radius zero. The 
two tangents drawn from the focus form, therefore, a conic 
whose equation has the character of the equation of a circle: 
the coefficient of xy is zero, the coefficients of «° and y? are equal. 

From this follows a new method for determining the foci of 
a conic, a method which one can present under the one or the 
other of the following forms: 


U 


306 PLANE GEOMETRY. BOOK III. 


‘1° If one put 
y =(4a + BB + D)x+(Ba+ OB +E)y + Dat EB+F, 


the quadratic equation of the tangents drawn from the point 
(a, 8) to the conic f (a, v7) =0 is (§ 282) 


f(x, YF (@ B)— 7 = 0). 


In order to express that the point («, 8) is a focus, it is suffi- 
cient to express the condition that this equation has the charac- 
teristics of the equation of a circle: in rectangular co-ordinates, 
the coefficient of ay is zero, the coefficients of a and y’ are 
equal. One has thus the two equations 


(c) Bf (a, B)—(Aa + BB+ D) (Ba + CB + E)=9, 
Af (a, B)—(Aa + BB + D)’ = Cf(a, B)—(Ba+ CB + EY’, 


which determine « and ®. If « and B be regarded in these 
equations as the current co-ordinates, they represent two conics 
whose points of intersection are the foci: thus, when the given 
curve is an ellipse or a hyperbola, these two conics intersect 
in four distinct points at finite distances, which are the two 
real foci and the two imaginary foci of the curve. One notices 
that the elimination of f(a, 8) between the preceding equa- 
tions furnishes an equation which can be written in the abbre- 
viated form 


Bf, —f%,)—(A—O) Ff p=0 


This equation, which represents a conic passing through the 
foci, is the equation of the ensemble of the axis of the conic 
[§ 137, eq. (24)]. 

2° The investigation of the foci is simplified if one form 
the condition 


(d) au? + 2buv + cv? ++ 2du+2ev+f=), 
which expresses the condition that the straight line ua + vy 


4+1=0 is tangent to the given conic ($ 126). Let («, 8) bea 
focus, the axis being rectangular ;. the straight line 


y—B—i(e—«)=90 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 3807 


ought to be tangent to the curve. Whence, for this particular 


straight line one has 
t —1 

ff Biman PEK a == 

B — ta B—te 
Expressing the condition that these values of wu and v satisfy 
the condition (d), one has, on developing and replacing ? 


by — 1, 
(e) f£(@’— Bp’) -—2da+2eB+a—c+ 21 (faB —ex — df +b). 
The straight line 





y—Btif@a—a)=0 


being tangent to the conic, one gets a second condition which 
leads to the preceding by changing ¢ into —i?. Therefore, 
if the real or imaginary point (a, 8) is a focus, the coefficient 
of i and the term independent of 7 ought to be zero separately 
in the condition (c), and one has the two equations 


f (a — B’)—2da+2eB+a—c=), 
fa8 —ee—dB+b=0, 


already attained above [eq. (c)] in another form. 
In case the axes are oblique and include an angle 6, one 
expresses the condition that the two straight lines 


y —B=(cosO+ isin 6)(% — ct) 


are tangent to the conic. 


INVESTIGATION OF SECANTS COMMON TO Two CURVES 
OF THE SECOND DEGREE. 


285. We have seen that two curves of the second degree, 
S=0, 8;=0, have in general four points in common; through 
these four points, which we will suppose for the present dis- 
tinct, one can pass three pairs of straight lines. In case the 
curves are real, the common points are real, or conjugate 
imaginaries taken by pairs. There are three cases to consider: 

1° If the four common points a, b, c, d are real, the three 
couples of common secants are evidently real. 2° If the four 


308 PLANE GEOMETRY. BOOK IIt. 


points are imaginary and conjugate in pairs, for example 
a and b, cand d, the two straight lines ab and ed, which pass 
through two conjugate imaginary points, are real; but the 
other four straight lines are imaginary; because if one of them 
ac were real, the points a and ec where the straight line ae in- 
tersects the two real straight lines ab and cd would be real. 
The straight line bd which passes through the two points 
b and d which are respectively conjugates of the points a and ¢, 
is conjugate to ac; similarly the straight line ad is the conju- 
gate of bc; thus, in this case, one has a couple of real secants 
ab and ed, and two pairs ac and bd, and ad and bc, each formed 
by two conjugate imaginary straight lines. 3° In case two of 
the points of intersection a and d are real, the other two cand d 
conjugate imaginaries, the two straight lines ab and cd are 
still real, and the other four imaginary; but the two imaginary 
straight lines of the same pair are not conjugates; because 
one knows that one imaginary straight line has but one real 
point, which belongs also to the conjugate straight line; the 
two imaginary straight lines ac and bd passing through two 
real distinct points a and 6 are not conjugates. 


286. The investigation of the points of intersection of the 
two curves depends on the solution of an equation of the 
fourth degree; but the question can be reduced to the solution 
of an equation of the third degree. The equation S +AS, =0, 
in which the parameter A is arbitrary, representing’ every curve 
of the second degree which passes through the points common 
to the first two, one can determine the parameter A so that this | 
equation represents two straight lines; since the two curves 
have three pairs of common secants, the value of A will be 
given by an equation of the third degree. 

Let 


if Aa? + 2 Bay + Cy? +2 Due+2Ey+ F=0, 
&) ee eer 
be the equations of the two curves. The new equation will be 


(17) (A+AA) a? +2 (B+ AB) xy =0; 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 309 


in order that it represent two straight lines, it 1s necessary 
and sufficient that the discriminant be zero (§ 124) and con- 
sequently that the constant A satisfies the equation of the 
third degree 


(A +AA)(C +0) (F + AF")—(A + AA)(E + AE")? 
—(CO + d0")(D + AD')? —(F + AF")(B + AB'P 
+2(B+)B)(D+ AD)(E + AE)=0. 


On arranging this equation with respect to A, we ##H have 
an equation of the form 


(18) A+@\ + OA? + AN = 0, 


where A and A’ are the discriminants of the curves S and S, 
and where © and @’ have the values 


@= A4'a+2Bd4+C'c4+2D44+2 H'e+ F', 
@' = Aa'+ 2 Bb'+ Cc! +2 Da'+ 2 Ee' + Ff’, 


a, b,c, d, e, f being the quantities already used above (§ 124) 
and a', b’, c’, d’, e', f' the quantities analogously formed with 
the coefficients of the conic S}. 

One real value of d gives two real straight lines, provided it 
makes the quantity 


(A +AA)(C+AC)—(B+ ABP 


negative, and two conjugate imaginary straight lines, in case 
it makes this quantity positive; because the first member of 
equation (17) has real coefficients and it decomposes into a 
product of two polynomials of the first degree, of which the 
coefficients are, in the first case, real, in the second case con- 
jugate imaginaries (§ 123). 

One imaginary value of A gives two non-conjugate imaginary 
straight lines. In fact, two real straight lines or two conjugate 
imagihary straight lines are represented by an equation of the 
second degree, 


(19) Alla? +2 B"xy = O"y? +2D"2+ E"y+ F"=0, 


310 PLANE GEOMETRY. BOOK III. 


with real coefficients. Equations (17) and (19), representing 
the same curve, have proportional coefficients; and since A is 
an imaginary quantity, one deduces 


Ale eG) So ene 
ACR e0 Go 2 ee 
The two equations (16) are identical. 

Suppose that the three roots of equation (18) are unequal. 
The three couples of straight lnes being distinct, the two 
curves have four distinct points in common. Owing to what 
has been said above, in case the three roots are real, the four 
points are all real or all imaginary; in case one root only is 
real, two points are real and two imaginary. In order to dis- 
tinguish the first two cases, one examines if three roots or one 
only make the quantity —f positive; in the first case, the 
four points are real, in the second they are imaginary. 


287. We have supposed thus far that the four common points 
are distinct. If the two points a and b coincide, the other 
two being distinct, the two curves are tangent at the real point 
a; the couple (ab, cd) is composed of the tangent at a which 
is real, and the real straight line; the other two couples (ac, bd), 
(bc, bd) coincide. The equation of the third degree has, there- 
fore, one single root and one double root, both real; the first 
gives the two real straight lines ab and cd, the second gives 
two straight lines, real or conjugate imaginaries, according as 
the two points ¢ and d are real or imaginary. 

Suppose that the two points a and e coincide, also the two 
points 6 and d; the two curves are tangents at the points a 
and b, which are real or conjugate imaginaries. One of the 
couples of straight lines consists of tangents at a and 6, which 
are real or conjugate imaginaries; the other two coincide with 
the double straight line ab, which is real. The equation of the 
third degree has, moreover, a single and a double root, both 
real; the first furnishes the two tangents, the second the chord 
of contact. (See, for a complete discussion, Chapter XII.) 


288. In order to give an application of what precedes, let 
us consider two ellipses having a common focus. These two 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. $11 


ellipses cannot-intersect in more than two real points; be- 
cause, after what has been said in § 260, the two ellipses which 
have a common focus and three common points coincide; they 
can have therefore but two real common secants. 
Let (a— a +(y—pyY—kyY =9, 
@—o)'-y — AY —khy* =; 
be the equations of two ellipses; since the two real common 
secants ky = + k'y' pass through the point of intersection I of 
the directrices D'I, DI (Fig. 166), it is easy to determine them 
geometrically. Suppose that 
the two ellipses intersect in 
two real points A and B; 
one of the real common se- 
cants is the straight line AB 
which passes through these 


two points; the other IL 
does not intersect the curves. 
In order to determine this 


second straight line, join the 
point A to the focus F and 
drop from this point the perpendiculars AH, AH' upon the 






% 





Pee ee ee ee 








Fig. 166. 


directrices; one has ha i! = and, consequently, 
! 
= ae Prolong the perpendicular AZ till EH is equal to 


AE; through the point H draw HL parallel to the first direc- 
trix, and through the point A, AZ parallel to the second direc- 
trix; the point of intersection Z of these two parallels will 
belong to the second real common secant JL. 


289. A circle intersects a curve of the second degree in four 
real or imaginary points; let 
(a —a)?+ (y—by—7r?=0 
be the equation of the circle, «= 0, 8 =0 those of a pair of 


real common secants; the equation of the curve of the second 
degree can always be written in the form 


(a—a)?+(y— bP? —r=kaB. 


812 PLANE GEOMETRY. BOOK III. 


The first member represents the square of the length of the 
tangent drawn from any point of the curve to the circle; 
whence follows the theorem: A circle being placed in any man- 
ner whatever in the plane of a curve of the second degree, the tan- 
gent drawn from every point of the curve to the circle is to the 
mean proportional between the distances of this point from the 
two real common secants in a constant ratio. 

Suppose that the circle be tangent to the curve in two real 
or conjugate imaginary points, the chord of contacts will be 
real, and the equation of the curve will take the form 


(vw — a)’ + (y — by — 7 = ke’. 


Thus, in case a circle is doubly tangent to a curve of the second 
degree, the tangent drawn from any point of the curve to the circle 
is to the distance of this point from the chord of contacts in a con- 
stant ratio. ‘The focus of a curve of the second degree can be 
considered as a circle with a radius zero, which has with the 
curve a double imaginary contact; the directrix is the chord of 
contact. 


290. By aid of the preceding theory one determines in a 
very simple manner the num- 
ber of normals that can be 
drawn from a given point to 
a curve of the second. degree. 
Let, for example, an‘ellipse be 
defined by the equation 





(1) Ba? + a®y? = a?b’, 


and P a point whose co-ordi- 
nates are ay, (Fig. 167). Let 
«and y be the co-ordinates of 
the foot M of one of the nor- 

oe mals; these unknown quanti- 
ties should satisfy equation (1) and also the equation 








I 


(2) W-y= pin (Mh — x) or Cay + by, — aay = 9, 


CHAP. IX. DETERMINATION OF CONIC SECTIONS 318 


which expresses the condition that the normal at the point M 
passes through the point P. It follows from the preceding 
that the point M is determined by the intersection of the 
ellipse (1) and of the equilateral hyperbola defined by equa- 
tion (2); one of the branches of the hyperbola passes through 
the center of the ellipse, and the two curves have at least two 
real points in common. The equation of the third degree on 
which depends the investigation of secants common to two 
curves, 1s 


(3) 4 a7b?3 + (a?a? + By? — A)A — Cay, = 0. 


If equation (3) have one real root, we have seen (§ 286) that 
the curves (1) and (2) cannot have more than two real points 
in common; if equation (3) have its three roots real, the curves 
(1) and (2), having at least two real common points, intersect 
in four real points. One ought to have, in the first case, 


(4) (a’x? + b*y,? — c*)? + 27 a’b?ctn’y? > 0, 

and in the second case, 

(5) (a°a,? + by? — cf)? + 27 a*b?cta?y)? < 0. 

If the co-ordinates x, y, satisfy the relation 

(6) (ax? + by? — cf)? + 27 a*b’ctx,y? = 0, 

the roots of equation (3) are still real, but it has one double 
root, and but three distinct normals can be drawn from the 
point P. The points P which satisfy this condition constitute 


a curve CDC'D' which has four cusps C, C’, D, D'. Equation 
(6) takes the very simple form 


re ae 4 
ata s + b3y3 = 3; 


it is plain that for every point within this curve, relation (5) 1s 
satisfied; that is, that through this point four real normals can 
be drawn, but no more than two real normals can be drawn 
through any point lying without this curve. 


314 PLANE GEOMETRY. BOOK IIt. 


EXERCISES. 


1. Construct a curve of the second degree, being given the 
directrix and three of its points. 

2. Construct a parabola, the focus and two of its points 
being given, or a point and a tangent. 

3. Construct a parabola, when its directrix and two of its 
points are given. 

4. Construct a hyperbola, if three of its points and the 
directions of the asymptotes be given. 

5. Construct a hyperbola, when an asymptote, a vertex, 
and one of its points are given. 

6. Find the locus of a vertex of a parabola which has a 
given focus and is tangent to a given straight line. 

7. Find the locus of the focus of a parabola whose vertex 
is at a given point and which touches a given straight line. 

g. Find the locus of the foci of curves of the second degree 
inscribed in a given parallelogram. 

9, A chord revolves about one of the foci of a curve of the 
second degree; find the locus of the point of intersection of 
the normals drawn to the curve through its two extremities. 

10. Two curves of the second degree have a common focus 
and an angle of constant magnitude which revolves about its 
vertex situated at the common focus; find the locus of the 
point of intersection of the tangents drawn respectively to the 
two curves at the points where they are intersected hy the sides 
of the angle. 

11. Find the locus of the point of intersection of the straight 
lines drawn parallel to two fixed directions through the extremi- 
ties of a chord of given length inscribed in a given circum fer- 
ence. 

12. Find the locus of the center of an equilateral hyperbola 
circumscribed about a given triangle. | 

13. Find the locus of the foci or the vertices of a hyperbola, 
having an asymptote and a directrix given. 

14. Find the locus of the centers of curves of the second 
degree which passes through the four points of intersection of 
two given conics. This locus does not change when each of 


CHAP. IX. DETERMINATION OF CONIC SECTIONS. 315 


the conics varies, remaining similar and concentric to the 
other, 

15. A variable circle touches a given ellipse at a given 
point; find the locus of the point of intersection of the tan- 
gents common to the two curves. 

16. Find the locus of the center of a hyperbola which has 
a given focus and which intersects in a given point a given 
straight line parallel to one of the asymptotes. 

17. Find the locus of the focus of a parabola which touches 
two given straight lines, one of them in a fixed point, the other 
in a variable point. 

18. Find the locus of the point of intersection of two parab- 
olas, which have a given point as focus, which touch a given 
straight line and which intersect at a given angle. 

19. Being given three points A, B, C, and an indefinite 
straight line, a variable segment MN is taken on this straight 
line, and is viewed from the point A at a constant angle; find 
the locus of the point of intersection of the two straight lines 
BM and CM. 

20. Two angles of constant magnitude revolve about their 
vertices placed at the extremities of the major axis of an 
ellipse; the point of intersection of two of the sides describes 
an ellipse; find the locus of the point of intersection of the 
other two sides. 

21. Find the locus of the vertices of an equilateral hyper- 
bola passing through a given point and having a given straight 
line as an asymptote. 

22. Being given a system of conics having the foci F and F", 
and a fixed straight line passing through the focus F; the tan- 
gents to these various conics, at points where each of them is 
intersected by this straight line, are tangents to the same 
parabola, whose focus is the point ’, and whose directrix is 
the secant. 

The portion of each tangent comprised between the conic 
and the parabola is viewed from the focus ¥”’ at a constant 
angle. 


316 | PLANE GEOMETRY. BOOK III. 


CHAPTER x 
THEORY OF POLES AND OF POLARS. 


291. Let us consider an algebraic equation of the degree m, 


S(@, y) aa 0, 


written in an integral form. The tangent, at the point whose 
co-ordinates are x and y, is represented by the equation 


(1) (X—a)/'.+(¥—ns'=0, 
or Xf. + ¥f',— (af ty’) =0. 


This equation involves, moreover, the co-ordinates of the point 
of contact to the degree m; but one can, owing to the relation 
(1), cause the terms of the mth degree to disappear. This 
reduction is easily accomplished by means of a special nota- 
tion, which we shall presently learn. Suppose that in equa- 


tion (1) # and y be replaced by and g and that every term be 


multiplied by 2”, the polynomial f(a, y) is transformed into a 
homogeneous polynomial of the mth degree with respect to 
the three letters 2, y, 2, a polynomial which we represent ly 
f(x, y, 2). tis evident that, if one put z= 1 in the last poly- 
nomial, one will get the given polynomial f(a, y). Itis known 
that in case a function f(a, y, z) be homogeneous and of the 
degree m with respect to the three letters x, y, 2, one has 
identically 
af, + uf + fi. = mF Y 2): 

Whence one has 


af! + yf', = mf (a, Yy, 2) — oe 


The value of the second member, when one puts z = 1, is equal 
to the quantity a’, + yf", which occurs in the equation of the 


CHAP. X. THEORY OF POLES AND POLARS. 317 


tangent; but the point of contact being on the curve, the first 
term mf(a, y, 2) reduces to zero; the expression af", + yf 1s 
therefore equal to the value which zf', takes when one puts 
z—1. One can thus put the equation of the tangent under 
the form 
Xf', + Yf', + of, = 0. 

For the sake of symmetry, one writes 
(2) Xf',+ Yi, +Zf'.=0. 

When one has taken the three partial derivatives of the 


homogeneous function f(a, y, 2), one replaces in equation 
(2) z and Z by unity. 


292. We propose now to draw from a given point p, whose 
co-ordinates are 2, and y,, tangents to the given curve. Call a 
and y the co-ordinates of one of the points of contact; since 
the tangent at this point passes through the point p, its equa- 
tion (2) will be satisfied by the co-ordinates x, and y, of the 
point p, which furnishes the relation 


att nt + Zf', = 9%, 
which, for the sake of symmetry, one writes in the form 
(3) nf, + ty tat, =9, 
in which one may replace at will z and z, by unity. The points 
of contact will be determined by the two simultaneous equa- 
tions (1) and (3). Since one of these equations is of the degree 
m and the other of the degree m — 1, the number of solutions 
will be at most m(m—1). Hence from the point p one can 
draw at most m(m — 1) tangents, real or imaginary, to a curve 
of the degree m. 

In case the curve is of the second degree, equation (3) is of 
the first degree, and one has two solutions, which are real or 
conjugate imaginaries. When the two solutions are real, one 
can draw from the point p to the curve two real tangents. In 
case the two solutions are conjugate imaginaries, the two tan- 
gents are conjugate imaginaries, but the chord of contact (3) 
remains real. The general equation of curves of the second 
degree tangent to a curve of the second degree represented by 


9 


318 PLANE GEOMETRY. BOOK III. 


the equation f(x,y) =0,at points where it is intersected EO 
the straight line (8), is (§ 281) 


Sf (&, i r (tf. + nS’, - Sy — 0, 
d designating an arbitrary parameter. If A be determined in 
such a way that this curve pass through the point 2, y,, it will 
reduce necessarily to the system of two tangents emanating 
from this point which are represented by the equation 


4 f(y mS @ N- CS + nS’, tas) = 0 


HARMONIC PROPORTION. 


293. Being given two points A and B, one knows that there 
exists on the straight line AB two points C and D so situated 
that the ratio of their distances from the two points A and 5 
is equal to a particular given ratio (Fig. 168). ‘These two points 

C and D are called har- 

A Le CB D monic conjugates with re- 

Fig. 168. spect to the two points 

A and B. It follows 

that there is an infinity of systems of harmonic conjugate 

points with respect to two given points; one can choose one 

of the points at will. In case the point C approaches the 

mid-point O of the straight line AB, the conjugate point D 
moves toward infinity, and conversely. 

We represent by the symbol AB the distance of the point A 
from the point B, affected with + sign or — sign, according as 
the point B is to the right or left of the point A. In accord- 





ance with this convention it follows that AB =— BA, and the 
position of the points C and D is expressed by the relation 
ye LO Nae O 
©) BC BD 
This relation can be written in the form 
CA___ CB, 
Dae ps 


one sees that, conversely, the two points are harmonic conju- 
gates with respect to the two points C and D. 


CHAP. X. THEORY OF POLES AND POLARS. 319 


If the relative positions of the four points be determined by 
the distances of one of them from the other three, the preced- 
ing relation becomes 


2 
©) AB =a¢' ap 
On reckoning the distances from the point O, the mid-point 


of AB, one has 
(6) OC. OD= OB. 


294. Turorem I.— Being given a conic section, if through a 
point p of the plane one draws any secant mm' (Fig. 169), the 
locus of the point p', the harmonic conju- 
gate of p, with respect to the two points = \>~-~___ 
of intersection m and m' of the secant oo 
with the curve, is a straight line. Let 


Sf (@, y) = Av? + 2 Bry + Cy? + 2 Dx Bo 






; Fig. 169. 
be the equation of the curve, a, and % i 


be the co-ordinates of the point p; any secant drawn through 
the point p will be represented by the equations 


7 RE os Ue ES 
(7) a b P» 





in which a and b are two constants, and p the distance of the 
point p from any point m of the straight line, affected by the 
+ sign or — sign, according as the point m is on the one side or 
the other of the point p; whence it follows that «= 2, + ap, 
y=%+ bp. On substituting these values in the equation of 
the curve, an equation of the second degree in p is found, 


FS (@ + ap, ¥, + bp) oes 


which determines the distances p’ and p" of the point p from 
the two points m and m'. The equation developed becomes 


(Aa? + 2 Bab + Cb’) p? + (af',, + Of"y,) p +S @y m1) = 9; 


320 PLANE GEOMETRY. BOOK ILI. 
or, if : be regarded as unknown, 
p 


Fey Wat O',+ if, b+ (Ae +2 Bab + CH) =0. 


Call r the distance of the point p from its harmonic conjugate 
p'. Owing to solution (5), one should have 


/ Tee eh En | 
, (oy 


pp 
But, by virtue of the last equation, 








Die eae OF y,. 
eed ST (@y I) 
whence 
2. of. +O, 
ae I (@y, ") ‘ 
or Od a0 yt Zhe, iy). 


The point p' belongs to the straight line pmm', and the co-ordi- 
nates « and y satisfy equations (7) of this straight line, that is, 
one has x—a=ar, y—y,=br; on replacing ar and br by these 
values in the preceding equation, the variable parameter a and 
b will be eliminated, and one gets the equation of the locus 


(8) (w — a) Fs, oO aes UA ere +2 f(x, %) = 9, 


which is of the first degree. Thus the locus sought is a 
straight line; this straight line P is called the polar of the 
point p and the point p the pole of the straight line P. 

By calculation it follows that the constant term 


2 f (a; N)— S's, bee CP * 
reduces to 2Dr,+2Ey%,+2F and the preceding equation 
becomes af! + of'y, + (2 Da + 2Ey,+2F)=090. 


This reduction can be made in another manner; suppose, as 
above, that in the polynomial f(x, y), # and y be replaced by 


CHAP. X THEORY OF POLES AND POLARS. 321 


* and 4 and that all the terms be multiplied by 2’; this poly- 
z z 


nomial will be changed into a homogeneous polynomial of the 
second degree, which we will represent by f(#, y, 2). By 
reason of the theorem of homogeneous functions which has 
been used in § 291, one has the identity 


af", + yf! + 2f', = 2h (a, Y; Z) 5 
whence it follows 
2 f(a, Y; z) rs af", — ae _ af 
or, on replacing «, y, 2 by %, %y 1, 
2f(@y Yy %1) — af 's, a Ss = “S's: 
Equation (8) can therefore be put under the form 
af'. + uy, tasis, =9% 
or, more symmetrically, 
(9) af . — yf . a af - — 0, 
in which, after the derivatives have been constructed, z and 2, 
are replaced by unity. If this equation be developed, one sees 
that it is not changed when the letters # and x, y and y, 2 and 
z, are permuted, and thus one obtains equation (3) of the chord 


of contact. Hence the polar of the point p coincides with the 
chord of contact with respect to this point. 


295. Next we examine the relative positions of the pole and 
of the polar. Through the point p draw a secant mm' (Fig. 
170) parallel to the chords which 
the diameter passing through the 
point p bisects; the point p being 
the mid-point of mm’, its har- 
monic conjugate is at infinity on 
this secant; whence one deduces 
that the polar P is parallel to the 
chord mm', that is, to the direc- 
tion conjugate to the diameter = + 
passing through the point p. 

Let o be the center of the curve and p!a point of the polar 


x 





Fig. 170. 


32% PLANE GEOMETRY. BOOK It. 


situated on the diameter op, that is, the harmonic conjugate 
point of the point p with respect to the two extremities c and 
c' of the diameter, then one has op-op'=oc’. If the pole p 
be moved along the diameter oc, the polar P will be moved 
parallel to itself; when the pole moves from o to c¢, the polar, 
in the first place situated at infinity, moves toward the curve 
and becomes tangent to it at c; if the pole crosses the curve 
and continues its motion toward infinity, the polar intersects 
the curve in two real points and moves toward the center of 
the curve. 

In case the curve is a parabola, the point c’ being situated 
at infinity, the point ¢ is the mid-point of pp’. 

It is easily seen that the converse is true: every straight 
line has one pole and only one, except in the case of the 
parabola, when the straight line is parallel to the axis. The 
curve being referred to any axes, in order to determine 
the co-ordinates x, and y, of the pole p of a given straight 
line wa + vy + w= 0, it will be sufficient to identify this equa-. 
tion with equation (9), which represents the polar of p, which 
furnishes the two relations, 


(10) Le ee 


On calling : the common value of these ratios and devel- 


oping, one gets 

Anda, -l- Bry, -+ Dx =U, ae 

BrA2x, + Ody, -+- EX = Vy 

DdX2x, +- Ey, + FX = W, 
equations of the first degree in Aa, Ay}, A. If the conic does 
not reduce to two straight lines, the determinant A of the coeffi- 
cients of the unknown quantities is not zero, and on using the 


general formulas for the solution of equations of the first 


degree, one has 
Ada, = aw + bu + dw, 


Ady; = bu + cv + ew, 
AX =du+evu+ fu. 


CHAP. X. THEORY OF POLES AND POLARS. 823 


In case the value found for d is not zero, these equations deter- 
mine a, and y,. The value of 4 is zero in case the coefficients 
of the straight line satisfy the condition 


du + ev + fw = 0, 


that is, on supposing f different from zero, when the straight 
line passes through the center of the conic whose co-ordinates 


are ° fi or, supposing f to be zero, the case of the parabola, 


when the straight line is parallel to the axis of the parabola. 
Remark. —If the curve of the second degree reduce to two 
distinct straight lines whose equations are 


a=le+my+n=0, B=le+m'y+n'=), 
one will have identically 
F (@, y) = oB; 
S,.=(B+Ua, fi =mB+m'a, fl, =nB+n'e; 
and on putting | 
qj =ley+mytn, B= le +my,+n', 


the polar of the point with the co-ordinates 2, y, will have the 
equation 
wf", + nt’, +f", a Ba, am ap, = 0, 


the equation of a straight line passing through the point of in- 
tersection of the two lines « = 0, 8B = 0, or parallel to these two 
straight lines in the case where they are parallel to each other. 
It follows from this form of the equation of the polar that: 
The polar of any point of the plane other than the point of 
intersection of two straight lines passes through this point of 
intersection. The polar of the point of intersection is inde- 
terminate. If the pole describe a straight line B—ma=0, 
passing through the point of intersection of the two straight 
lines, the polar remains fixed, its equation being B + me = 0. 
Conversely, a straight line which does not pass through the 
point of intersection has this point for pole. .A straight line 
6B +ma= 0, passing through the point of intersection of two 


824 PLANE GEOMETRY. BOOK III. 


straight lines, has an infinity of poles situated on the straight 
line B—ma=0. (See § 103.) 


296. THreorEeM II. — The polars of all the points of a straight 
line pass through the pole of this straight line, and, conversely, 

the poles of all the straight lines which 

pass through the same point are situated 
p on the polar of this point. 


a On the straight line P whose pole is 
p, select any point gq (Fig. 171); the 


straight line pq intersects the conic in 
two points m and m'; these two points p 
and gq being harmonic conjugates with 
respect to the two points m and m', the 
polar Q of the point g passes through the 
point p. 

Conversely, let g be the pole of any straight line Q passing 
through the point p; the two points q and p being harmonic 
conjugates with respect to the two points m and m'in which 
the straight line pq intersects the conic, the point g belongs to 
the polar P of the point p.. 


m’ 


Fig. 171. 


ConsuGATE StraicHtT Lines. — Two straight lines are said 
to be conjugates with respect to a conic in case the pole of 
either lies on the other. Let 


o 


ux + vy + w=), wet vy +w'=0 


be two conjugate straight lines. 
On expressing the condition that the pole (a, 4) of the first 
lies on the second, it follows that 


u'z, tv'y, + w'=9, 
or, writing the preceding values for a and 4, 
u' (au + bv + dw) + v' (bu + cv + ew) + w! (du + ev + fw)= 0, 
auu' + b(wo! + vu')+ cov! + d (uw! + wu’) 
+e(vw! + we')+ fww'=0. 


CHAP. X. THEORY OF POLES AND POLARS. 325 


This condition can, moreover, be written as follows, if one put 
d (u, v, w)= au® + 2 buv + cv? + 2 duw + 2 evw + fw’, 
ulp!, + lb, + w'p', = 0, 
ud', + vd', + wl, = 90. 


297. Turorem III. — Being given a conic section, if any two 
secants pmm', pni', which intersect 
the curve in m, m', n, n', be drawn 
through a point p (Fig. 172), the 
points of intersection g and q' of the 
straight lines mm, m'n', or m'n, mn’, 
belong to the polar of the point p. 

We remark in the first place : 
that Theorem I. holds, in case the FG" 
locus of the second degree reduces ‘ss . 
to a system of two straight lines; ; rs mi 
in this case the polar of the point p See ea 
passes through the vertex of the ares 
angle; for, if the secant which 
passes through the vertex be considered, the two points m and 
m! coincide with this point, p and p' being harmonic conjugates. 

This being established, let us consider the system of two 
straight lines, mn, m'n', which intersect in g. The straight 
line pmm' intersects the conic section and the two sides of the 
angle mgm! in the same points, m and m'; the point p’, a har- 
monic conjugate of the point p, is the same point on the secant 
pmm!' whether one regard this secant as belonging to the conic 
section or to the angle. The point p", a harmonic conju- 
gate of p, will remain the same, in both cases, on the secant 
pnn'. The polars of the point p, with respect to the conic sec- 
tion and to the angle, having two common points p’ and p", 
coincide; but one knows that the polar with respect to the 
angle passes through the vertex g, therefore the point q 
belongs to the polar of the point p with respect to the curve. 
For the same reason, the point q' belongs to this same 
polar. | 

CoroLttAry.—The curve being traced, one gets from this 








326 PLANE GEOMETRY. BOOK III. 


theorem the means for constructing the polar of the point p. 
Draw through the point p two secants, pmm!', pnn', by means 
of which two points, g and q', of the polar are determined. 

If the point p lie without the curve, the polar intersects the 
curve in two points, which are the points of contact of the 
tangents drawn from the point p. 

Remark.—In Fig. 172 the polar of the point q is the 
straight line pq', and the polar of the point q' is the straight 
line pg. The triangle whose vertices are the points p, gq, q', 
possesses, therefore, this remarkable property, that each of its 
sides is the polar of the opposite vertex with respect to the 
curve of the second degree. It is said that such a triangle 
is autopolar or 1s a conjugate triangle with respect to the curve 
of the second degree. Conversely, it is also said that the 
curve is conjugate to the triangle. 


RECIPROCAL POLAR FIGURES. 


298. Being given a plane figure composed of the points a, 
b,c, +++ and of straight lines A, B, C, .--, if one construct the 








Fig. 173. 


polars A’, B', C', --- of the points, and the poles a’, b’, c’, «- 
of the straight lines, with respect to a definite conic section, 
one forms a second figure composed, like the first, of straight 


CHAP. X. THEORY OF POLES AND POLARS. 327 


lines and of points. On treating the second figure in the 
same manner, that is, on taking the poles of the straight lines 
and the polars of the points, one gets the first figure. These 
two figures have been called for this reason reciprocal polar 
figures (Fig. 173). 

The straight line ab, which joins the two points a and 6 of 
one of the figures, has as pole the point of intersection of the 
two straight lines A’ and B' of the other figure; and con- 
versely, the point of intersection of the two straight lines A’ 
and B' of one of the figures has as polar the straight line ab 
of the other figure. If several points a, 6, ¢, +++ lle on a 
straight line in one of the figures, the straight lines A', B’, 
C', --- of the other figure pass through the same point, which 
is the pole of the straight line. Conversely, if several straight 
lines A, B, C, --- pass through the same point in one of the 
figures, the points a’, b’, c', --- of the other figure le in a 
straight line. 

A plane curve S being given, draw a tangent A to this curve 
and determine the pole a! of this tangent (Fig. 174). If the 
tangent A revolves about the 
curve S, the pole a! will de- 
scribe another curve S'. Let 
A and B be two tangents to 
the curve S, a’ and 6’ their 
poles; the point of intersec- 
tion m of the two straight 
lines A and B is the pole of 
the straight line a’b'. If the 
tangent B approach the tan- 
gent A as its limit, the point 
mm will approach the point of contact a of the tangent A; at 
the same time the secant a'b’ revolves about the point a’ and 
becomes tangent to the curve S' at the point a’. Hence, con- 
versely, the curve S is the locus of the pole a of a movable 
tangent A! of the curve S'. The points a and a! correspond 
to each other in such a way that the tangent at one of these 
points is the polar of the other. The two curves S and S!' are 
for this reason called reciprocal polars. 





Fig. 174. 


328 PLANE GEOMETRY. BOOK ITI. 


Let 
(11) BEM eat 
be the equation of an algebraic curve S of the degree m; the 


tangent A at the point a, whose co-ordinates are # and y, is rep- 
resented by the equation 


(12) XF',+ YF',+ ZF',=0. 


Call x, and y, the co-ordinates of the pole a’ of the straight line 
A, with respect to a curve of reference of the second degree 
J (a, y)=0; the equation of the polar of the point a! is 


(13) >. 4) 2 ts a7 - ae As = 0. 


The two equations (12) and (13), which represent the same 
straight line, should be identical, and one has the relations 


! ! ! 

14 ins PE dich 4 eatin 
( ) ee Vas Yap 
If, among the three equations (11) and (14), 2 and y be elimi- 
nated, one has the equation of the curve S,, the locus of the 
point a’. 

Seek, for example, the reciprocal polar curve of the conic 
section Aa? + By’? —1=0, with respect to the circle of refer- 
ence 





a+ y?—1=0. 
If wand y be replaced by “and “ these two equations assume 
z Zz 


the homogeneous forms A2’+ By?—2=0, a+ y—2#=0, 

and equations (14) become =. = = =" ; whence it fol- 

lowe: on puuing. 2-2; — 1, n=, y =43; by substituting 

these values in the equation of the given curve, one has the 
2 2 

equation =I a a —1=0. The polar reciprocal curve is a new 


conic section. 
299. The degree or order of an algebraic curve has been 


called the degree of the equation by which it is represented in 
rectilinear co-ordinates, or the number of points, real or imagi- 


CHAP. X. THEORY OF POLES AND POLARS. 329 


nary, in which the curve is intersected by any straight line. 
In like manner, the class of the curve is the number of tan- 
gents, real or imaginary, which can be drawn to the curve 
from any point of the plane. It is known that from any point 
two tangents can be drawn to a curve of the second degree ; 
curves of the second order belong therefore to the second class. 

It is very easily proven that two reciprocal polar curves S 
and §' (Fig. 175) are such that the order of one is equal to the 
class of the other. Any straight line P intersects the curve 
S in m points a, b,c, «++; to these a points correspond m 
straight lines A’, B', C', ---, tangents to the curve S', and pass- 
ing through the point p', the pole of the straight line P; con- 
versely, to each tangent A’ drawn from the point p' to the 


oY 


Gi 





Fig. 175. 


curve S' corresponds a point a belonging to the curve S and 
situated on the straight line P. . Hence, the number of 
tangents which can be drawn from the point p’ to the curve 
S' is equal to the number of points of intersection of the 
curve S by the straight line P, and, consequently, the class of 
the curve S’ is equal to the order of the curve S. Similarly, 
the order of the curve S! is equal to the class of the curve S. 

A curve of the second order being of the second class, it 
follows that the reciprocal polar curve of a curve of the second 
order is also of the second order. 

In like manner, it is easy in this case to determine the 
species of the curve. If the center o of the curve of refer- 
ence be situated without the curve S, two real tangents A and 


330 - PLANE GEOMETRY. BOOK IT. 


B can be drawn from this point to the curve S (Fig. 176); the 
poles of these tangents lying at infinity, it follows that the 
eurve S' has two _ infinite 
branches with different direc- 
tions; it is therefore a hyper- 
bola. Let a and b be the 
points of contact of the tan- 
gents A and B; the polars A’ 
and B', of these two points 
are tangents to the curve S’ 
at points situated at infinity ; 
they are, therefore, the asymp- 
totes. In case the center o of 
the curve of reference is situated on the curve S, the points a 
and b coincide with the point 0, the polar of this point, or the 
asymptote is removed to infinity, and the curve S' is a 
parabola. Finally, if the center o of the curve of reference be 
situated within, the curve S' is an ellipse. 

‘To any two points « and b of the conic S and the tangents 
A and B at these two points correspond two tangénts A’ and 
B' to the conic 8‘ and their points of contact a'and b!. To the 
point of intersection c of the straight lines A and B corre- 
sponds the straight line a‘b’, and to the straight line ab the 
point of intersection c' of the straight line A’ and B’. Hence, 
to a point ¢ and its polar ab in the first figure correspond in 
the second figure a straight line a‘b' and its pole cc’ « 


A’ 





Fig. 176. 


300. The method of reciprocal polars plays an important 
role in the study of conic sections; it is possible when a 
property of these curves has been found by the method of 
reciprocal polars to deduce immediately a correlative property. 
It has been demonstrated, for example, in § 275, that through 
five given points one can pass a conic section, and one only ; 
whence it follows that a conic section can be drawn tangent to 
five given straight lines, and one only. Imagine, in fact, that 
any conic section be drawn in a plane as the curve of refer- 
ence, and that with respect to this conic section one locates 
the poles a’, b’, c', d', e' of the five given straight lines A, B, 


CHAP. X. THEORY OF POLES AND POLARS. 831 


C, D, E; a conic section S' can be drawn through the five 
points a’, b', c', d', e'; the polar reciprocal curve of the curve 
S' will be a conic section S tangent to the five given straight 
lines. Conversely, to every conic section tangent to five 
straight lines corresponds a conic section passing through 
five points; since but one conic section can be passed through 
five points, it follows that but one conic section can be drawn 
tangent to five straight lines. 

Let.us consider the polars of a point p with respect to vari- 
ous conics which pass through four given points; if f(a, y)= 0, 
and F(x, y)=0, be the equations of two of them, the equation 
f+kF=0, in which k is an arbitrary parameter, will repre- 
sent the ensemble of these conics. The polar of the point p, 
whose co-ordinates are w and y, has the equation 


ay(f', + KF") +n(f, +k!) +20, + KF") = 0, 
or (af, =e n'y i S's) +k (eF", a is yt", a i 2, F",) == 0% 


all the polars pass through the point of intersection p!' of the 
two straight lines 


1S". Om WT", aa af". a 0, x Et", ee yl", me ZF", = 0. 


It is clear that, conversely, the polars of the point p', with 
respect to the various conics, all pass through the point p. 

If the figure be transformed by the method of reciprocal 
polars, it follows that the locus of the poles of a straight line 
P with respect to the various conics tangent to the four given 
straight lines is a straight line. If the straight line P be 
moved to infinity in an arbitrary direction, its pole with re- 
spect to each of the conics is the center of this conic; hence 
the locus of the centers of conics tangent to four given straight 
lines is a straight line. Each of the diagonals of the quadri- 
lateral formed by four straight lines can be regarded as an 
ellipse or a hyperbola infinitely flattened and tangent to the 
four straight lines; the mid-points of the three diagonals 
belong to the locus and determine this straight line (§ 73). 


301. THrorEmM IV. — Two conics are given in a plane: 
1° The points of intersection of three pairs of common secants 


9 


332 PLANE GEOMETRY. BOOK IIT. 


determine a triangle each vertex of which has as polar, with 
respect to each of the conics, the opposite side; 2° The points of 
intersection of the four common tangents to the two conics are 
situated two by two on the sides of this triangle. 


Let a, b, c, d (Fig. 177) be the four points common to the 
two conics; the points of intersection m,n, p of the three pairs 





Fig. 177. 


of common secants form a triangle mnp, each vertex of which, 
according to Theorem IILI., has as polar, with respect to each of 
the two conics, its opposite side. We notice that these three 
points are the only ones which enjoy the same polar property 
with respect to the two conics. Let m' be a point having the 
same polar with respect to the two conics; the straight line 
m'a intersects this polar in a certain point g and each of the 
conics in a new point which is the harmonic conjugate of a 
with respect to the two points m' and q; these two new points 
ought to coincide, the straight line m'a passes through one of 
the common points }, ¢, d, for example through the point 0. 
Then the straight line m'c will pass through the point d and 
the point m! wil! coincide with the point m. 

Imagine now that the preceding figure be transformed by 
the method of reciprocal polars. To the two conics there will 
correspond two other conics; to the points a, b, c,d common to 
the first two the tangents A’, B', C', D' common to the two new 


CHAP. X. THEORY OF POLES AND POLARS. S00 


conies, which shows that the two conics have four common 
tangents. 

Consider one of the tangents common to the two given 
conics; let g and g'be the points of contact and e the point 
where it intersects the straight line np; the straight lines mg, 
mg' intersect the conics in two other points h and h'; the point 
m, having the same polar np with respect to the two conics, the 
tangents at h and h' pass through the point e; the straight line 
np being also the polar of the point m with respect to the two 
angles geh, g'eh', the two straight lines eh, eh' coincide, and the 
straight line ehh' is a second common tangent. For the same 
reason, the point of intersection f of one of the other common 
tangents with the straight line np belongs to the fourth tangent. 
Thus the six points of intersection of the four common tan- 
gents are situated two by two on the sides of the triangle mnp. 

Whence it follows in addition to what precedes that the 
‘chords of contact pass four by four through the points m, 2, p. 


302. Let us study in particular the case where the curve of 
reference is a circle of radius 7; the polar A’ of a point a is 
perpendicular to oa and at a distance from the center equal 


to ”. The straight lines which join the center to the two 
oa 


points a and 6 inclose an angle aob 
equal to that included by the polars 
A'and B' of these points (Fig. 178). 
Through the center o draw lines 
parallel to the straight lines A’ and 
B'; from the points a and b draw 
lines perpendicular to these straight 
lines; the right-angled triangles oae, 
obf are similar and give the propor- 
tions 
oa _ae_ac+ce _ ac+ og, Fig. 178. 
ob bf bd+df bd+oh’ 








‘whence follows oa (bd + oh) = ob (ac + 0g); but one has 


2. 
oa: oh =ob-og=rT; 


3834 PLANE GEOMETRY. BOOK IIt. 


whence it follows that oa - bd = ob + ac, or = = i Hence the 
0 


distances of two points from the center are proportional to the 

distances of each of them from the polar of the other. 
Find the reciprocal polar of a circle of radius r' with respect 
to acircle o. Let C' be the polar of the center ¢ of the given 
circle (Fig. 179); draw to this circle 


, any tangent A and locate the pole a’ 
d of this straight line; owing to the 
| Ae preceding property, one has 
ON OG) Oa oa' oc 
te re a rs 
OG aid vy! 

the ratio of the distances of each of 
a the points a' of the locus from the 





point o and the fixed straight line C’ 
is constant; therefore this locus is a: 
curve of the second degree, of which the point 0 is one of the 
foci and the straight line C' the corresponding directrix. 

By means of this transformation, most of the focal proper- 
ties of curves of the second degree may be deduced from the 
properties of the circle. Thus, for example, two tangents A 
and B to the circle ¢ form equal angles with the chord of con- 
tact ab; to two straight lines A and B correspond two points 
a' and b! of the conic section; to the two points a and 6 of the 
circle correspond the tangents A' and B' to this canic section 
at the points a! and b'; to the straight line ab or M corre- 
sponds the point of intersection m! of the straight lines A‘ and 
B'. The radii vectores drawn from the focus o to the points a’, 
b', m' forming angles with one another equal to those of their 
polars A, B, M, it follows that the straight line om' is the 
bisector of the angle a'ob' (§ 255). 

The locus of the vertex m of a constant angle circumscribed 
about a circle is a concentric circle. To the two tangents A 
and B drawn from the point m to the circle correspond two 
points a’ and b' of the conic, and to the point m the straight 
line a'b'; the angle a’ob', being equal to that of the straight 
lines A and B, is also constant; since the point m describes a 


_ Fig. 179. 


CHAP. X. THEORY OF POLES AND POLARS. 335 


circle whose center is ¢, its polar a'b' envelopes a conic section, 
of which o is one of the foci and the polar of the center ¢ 
the corresponding directrix. Hence the chord viewed from a 
focus of a conic section and subtending a constant angle envel- 
opes a conic section which has the same focus and the same 
directrix. The chord ab of the circles envelopes a concentric 
circle; therefore the point of intersection of the tangents to 
the conic section at a! and b!' describes a conic section which 
has also the same focus and the same directrix. 


ENVELOPE CURVES. 


303. In what precedes we were led to consider curves which 
were tangent to a series of straight lines; in case a point 
describes a curve, its polar remains tangent to another curve. 
The envelope of a mov- 
able curve is the curve to 
which this hne remains 
constantly tangent. 

Let 


| Fig. 180. 

(1) f@y a=0 

be an equation involving a variable parameter a. To each 
value of a corresponds a definite curve. Give to the param- 
eter two consecutive values a and a+h; the curve (1) and the 
curve 

(2) S(@, y,a+h)=0 

intersect in a point M' (Fig. 180), whose co-ordinates satisfy at 
the same time equations (1) and (2). The system of these two 
equations can be replaced by the following: 


f (a, ¥, a) = 0, SI, Y,a+ hes Y; a) 0, 








which, when h approaches zero, reduces to 


(3) I (@, Y; a)= 0, F'a(@, Ys a= 0; 
hence, when h approaches zero, the point. M'is displaced on 
the curve (1) and approaches a limiting position M; it is this 


336 * PLANE GEOMETRY. BOOK III. 


limiting point which is represented by the system (3). Each 
of the curves (1) contains a limiting point; the locus of these 
points, which is sometimes designated by the name of the 
locus of the ultimate intersections of the curve represented by 
equation (1), is obtained by eliminating a between equa- 
tions (3). 

Consider the new system of equations (1) and (2), in which 
ais regarded as a variable and h as a constant; this system 
represents the locus of the points in which each curve (a) is 
intersected by the curve (a +h). Two of these points he on 
the curve a; namely, the point of intersection M' of the curves 
(a) and (a+h), the point of intersection M" of the curves 
(a —h) and (a). When h approaches zero, the points M ' and 
M" approach the same limiting position M, and the locus 
becomes tangent to the curve (a) at the point M. Hence, the 
locus of the ultimate intersections of the curves represented by 
equation (1) is tangent to each of these curves. 

Remark. — When f(a, y, @) is a polynomial with respect to 
a, to eliminate a between equations (3) is to express the condi- 
tion that the equation in a 


I (2, Ys a)= 0 
has a double root. 
For example, if a enters to the second degree in f (x, y, a), 
and if the equation of the movable curve have the form 


MG 4-2 Na + PF —0, 


M, N, P being polynomials in a and y, the equation of the 
envelope obtained by expressing the condition that the equa- 
tion in a has a double root will be 


N’?— MP=0. 
According to this method it can easily be verified that the 
envelope of the conics whose equation is 
era, 
ee ert 
when a, B, y represent given linear functions in « and y 


and A a variable parameter, is composed of four straight lines 
(§ 282. 2). 


CHAP. X. THEORY OF POLES AND POLARS. 337 


304. Assume now that the movable curve be represented by 
an equation 


(4) S (a, Y; Os; b)= 0, 


containing two variable parameters a and }, connected by the 
relation 


(5) (a, d)=0. 


If b’ be called the derivative of 6 considered as a function 
of a given by equation (5), one has ¢',+ $4b'=0; whence 
! 





b'——-—*. But if the derivative with respect to a of the 


function (®t, y, a, b) be equated to zero when 6 is regarded as 
a function of a, one has f',+/',b'=0; whence follows the 
relation 

! 
(6) ae 
and in order to find the equation of the envelope, the two 
parameters a and b are eliminated by means of the three equa- 
tions (4), (5), (6). 

ExampeE ].—F ind the envelope of the normal to a pa- 
rabola. The normal to the parabola 7°? = 2px, at the point 
M (Fig. 181) whose co-ordinates 
are a and y, has the equation 
p(¥—y)+y(X—2)=0; if aw be 


2 
replaced by its value J this equa- 
tion becomes mad 


3 
) Pet VD) 








it involves an arbitrary parameter 
y; it is necessary to equate to zero 
the derivative with respect to y, 


al e yf” 
(8) eae 
and to eliminate y from equations 
(7) and (8). On replacing y’ in Fig. 181. 
B's 








338 PLANE GEOMETRY. BOOK III. 
equation (7) by its value derived from ‘equation (8), one has 


oa acy 2 Oe 
2(X—p)’ 


substituting this value of y in equation (8), one obtains the 
equation of the envelope 
(9) Yy?= AC aa 2 

27 p 

This curve takes the form given in the figure; it has a cusp 
at CO, because the tangent at this point being normal to the 
parabola at the vertex A coincides with the axis AX. When 
the point M describes the branch AB of the parabola, the 
normal revolves about the branch CD of the envelope; and, 
similarly, when the point M describes the branch AB! of 
the parabola, the normal resolves about the branch CD’ of the 
envelope. 

If one wishes to draw the normals to the parabola from a 
given point P, it is sufficient to regard X and Y in equation 
(7) as co-ordinates of the point Pand the ordinate y of the foot 
M of the normal as unknown; three normals can be drawn 
from the point P to the parabola, or one only, according as this 
equation of the third degree in y has three real roots or one 
only. This problem is like that of drawing tangents from the 
point P to the envelope; hence the envelope, which is of 
the third degree, is also of the third class. In case'the point 
P is situated between the branches of the envelope, three 
tangents can be drawn from this point to the envelope, and, 
consequently, three normals to the parabola; but, in case the 
point P is situated at P’ without, one tangent only can be 
drawn to the envelope, and, consequently, one normal only to 
the parabola. 

Exampte II. — Let us find the envelope of the normals to 
the ellipse 


eee ys 
ae ee 


the equation of the normal at the point (2, ¥) 


CHAP. X. THEORY OF POLES AND POLARS. 339 


ex PY 
v y 





(10) (a? — b°)=0 


involves the variable parameters # and y connected by the 
relation 


(11) ~+2—1=0. 





& & 

a? b? 1 
Gk ney ba 
Te. Pp 


the third ratio has been found by adding the numerators and 
denominators, after having multiplied the two terms of the 
first by a, the two terms of the second by y, and using equa- 
tions (10) and (11); whence follow 


Substituting these values in equation (11), one obtains the 
equation of the envelope 


(12) CaleiColer 


This curve has four points of inflection (Fig. 182). When the 
foot M describes the are AB of the ellipse, the normal revolves 
about the are CD of the envelope. If one wishes to draw the 
normals to the ellipse from a given point P, whose co-ordinates 
are X and Y, the two simultaneous equations (10) and (11) 
will determine the co-ordinates # and y of the foot of each of 
the normals; the feet of the normals are the points of inter- 
section of the given ellipse (11) and of a hyperbola (10); hence 
there are four solutions. Moreover, this problem resolves itself 
into drawing the tangents from the point P to the envelope; 
it follows that the envelope which is of the sixth degree is of 


340 PLANE GEOMETRY.:- BOOK III. 


the fourth class. In case the point P be situated within the 
envelope, four tangents can be drawn from this point to the 
envelope, and, consequently, 
four real normals to the el- 
lipse; but if the point lie with- 
out, for example at P', two 
real tangents only can be 
drawn to the envelope, and, 
consequently, two normals to 
the ellipse; thus the results 
obtained in § 290 are proved 
anew. 

The envelope of the normals 
of ahyperbola has the equation 


Fig. 182. | (13) Gar pee (ae — 7. 
Cc Cc 


305. In case a variable plane moves in a fixed plane, it can 
happen that a curve CD of the variable remains tangent toa 
curve AB of the fixed plane; this second curve is the envelope 
cf the first. Let CD and C'D' (Fig. 183) be two consecutive 
positions of the variable curve, 
M' a point of intersection of these 
two curves. As the curve C'D! 
approaches continuously the curve 
CD, the point M' approaches a lim- 
iting position M, which is a point 
of the envelope (§ 303). Let M, 
be the point of the curve CD which 

nee, has arrived at M’, when this curve 
takes the position C'D'. We have seen (§ 31) that a variable 
curve can be brought from one position to another by rotating 
it about a fixed point J,; the perpendicular Ph, erected at the 
mid-point of the chord I,M', passes through the point f. 
But the two points M, and M' have the point M as their 
limiting position, and the straight line PJ, becomes the nor- 
mal common to the curve CD and its envelope at the point ; 
this normal passes therefore through the point J, the lmiting 














CHAP. -X. | ENVELOPE CURVES. 341 


position of the point 4, whence it follows that the normals to 
the various curves situated in the variable plane, at the points 
where they touch their envelopes, for one position of the variable 
plane, pass through the same point I. For the same position of 
the plane, this point is that point through which the normals 
to the curves described by the points of the variable plane 
pass. 

In case a curve of the variable plane remains tangent toa 
given curve of the fixed plane, the common normal to the two 
curves can be used to determine the point J. The polar of a 
curve AB (Fig. 31) with respect to the point O (§ 38) is none 
other than the locus described by the vertex P of a right 
angle OPM situated in the movable plane, which so moves 
that one of its sides PM remains tangent to the curve AB, 
while the other passes through the fixed point O; that is, 
remains tangent to a circle of radius zero whose center is O; 
whence follows the construction that the point J is found by 
the intersection of the perpendiculars drawn from the points 
Oand M to the two sides OP and PM of the right angle. (See 
construction given in § 38.) 


TANGENTIAL CO-ORDINATES. 


306. A curve may be regarded either as the locus of a point, 
or as the envelope of a variable straight line. From the second 
point of view, we represent the straight line by an equation 
of the form 


(14) ux+vy+1=90, 


and we say, by analogy, that the two parameters wu and »v, 
which determine its position, are the co-ordinates of the 
straight line. 

If one be given an equation 


(15) $(u, ¥) =0 


between these two parameters, and one of them be allowed 
to vary in a continuous manner, the other will also vary in 
general in a continuous manner, and the straight line will be 


$42 PLANE GEOMETRY. BOOK IIt. 


given a motion in the plane, enveloping a curve. One can 
think of equation (15) as representing a curve through a series 
of its tangents, by means of a new system of co-ordinates wu and 
v, which are called for convenience tangential co-ordinates. 

In order to get the equations of this curve in linear co-ordi- 
nates, it suffices, after what has been given in § 304, to elimi- 
nate wu and v from equations (14), (15), and ; 


a 
(16) roa 
If equation (15) be algebraic, the equation in a and y, the 


equation which we will deduce, will also be algebraic. Make 
equation @(u, v) =0 homogeneous by replacing wu and v by 
os “ and removing the denominators. The équation ¢=0 
can be replaced by the following 

ud', + Vb + $y = 9, 


in which one puts, after constructing the derivatives, w= 1. 
Therefore on calling A the common value of the ratios in (16), 
it is necessary to eliminate wu, v, A from the four equations 


v= AP! ¥ — r}',, 
ux+vy+1=0, 
ud, + vd', + >! = 9. 


Multiply the first of these equations by wu, the second by 2, 
and using the last two equations, we will have the equation 


1 si AP's 


which can replace one of the last two equations, for example 
the last. One will have therefore to eliminate uw, v, » from 
the equations 


(17) t= rAd |; ae r¢',, 1= AP's 
ux+ovy+1=0. 


The degree of equation (15), written in an integral form, 
indicates the class of the curve. For if a and y be the 


CHAP. X. TANGENTIAL CO-ORDINATES. 8438 


co-ordinates of any point of the plane, each system of values 
of u and v, satisfying the two equations 


UN + VY, +1=0, d(u, v) =), 


will determine a tangent, real or imaginary, passing through 
the point in question. In case equation (15) is of the second 
degree, the curve, being of the second class, is also of the 
second order (§$ 309). 

An equation of the first degree 


Au + Bu + C= 0, 


in tangential co-ordinates, represents a point, the point which 


has the linear co-ordinates 2 = a Yo= a3 because this equa- 
tion, put under the form 


UX + vy +1 =), 


indicates that the variable straight line passes always through 
the fixed point (a, Y); the envelope reduces therefore to a 
point. 

The properties of the equation of the first degree in linear 
co-ordinates, which has been studied in Book II., is here re- 
produced, with the modification that points are replaced by 
straight lines and straight lines by points. Thus the equation 


vo—v=atu—u"), 


in which the parameter @ is arbitrary (§ 64), is the general 
equation of points situated on the straight line (w', v’). The 
equation (§ 66) 





(18) vot =" ww! 


represents the point of intersection of the two straight lines 
(u', v'), (ull, v"). ae 
Consider two consecutive tangents of the curve (15), and sup- 
pose that the second approaches continually the first; their 
point of intersection, represented by equation (18), will have 


344 PLANE GEOMETRY. BOOK III. 


as a limit the point of contact of the first tangent; the point 
of contact is therefore represented by the equation (§ 89) 


Pv 
v—v=——*(u—w), 
p 





t 
a: 


or 
(19) (u—u')o'. +(v—v') dl, =0. 


On replacing u and v by “ and ~ , in order to make the equa- 


tion homogeneous (§ 291), this equation is simplified and 
written in the form 


(20) ug', ate vd, a wp! y = 0. 


307. It is well to notice that the investigation of the en- 
velope of a variable straight line can be reduced to the theory 
of reciprocal polars because this envelope curve is the recipro- 
cal polar curve of the curve S' described by the pole of the 
straight line, with respect to a given conic. If one choose as 
curve of reference the imaginary circle 2+ y?+1 = 0, and 
if one put a,=w, y=, the straight line aa, + yyz,t1=0, 
the polar of the point (a, y,) coincides with the variable 
straight line (14); hence the curve S' has the equation 
¢ (2, y,) =0 in linear co-ordinates. 


ExampPLe I.— Find the envelope of a straight line such that the prod- 
uct of its distances from two fixed points F and F’ be equal to a given 
constant quantity. On choosing the straight line FF’ as x-axis, and a 
perpendicular at the mid-point of this straight line as y-axis, calling 2c 
the distance FF’, 52 the constant product, and representing the variable 
straight line by the equation wx + vy + 1=0, one has the relation 


(c? + b7)u? + 622? —-1=0, 


connecting the two variable parameters u and v; it is necessary to choose 
the + sign or — sign, according as the straight line passes to the right 
or left of the two points, or between them. ‘The curve S', having the 
equation 

(c? + b?) x? ae b2y?, —l= 0, 
the equation of the curve sought S, or of the reciprocal polar (§ 298), is 


2 
ane Y_.—1=0, 
c2 + b2 + h2 








CHAP. X. TANGENTIAL CO-ORDINATES. 345 


This is an ellipse or a hyperbola whose foci are the points F and F’. 
This property is the converse of the theorem demonstrated in § 259. 

Exampve II. — Being given a quadrilateral abcd, find the envelope of 
a straight line such that the product of the distances of two opposite 
vertices of the quadrilateral from the straight line is to the product of 
the distances Of the other two vertices from the straight line in a constant 
ratio. Call a; and y;, x2 and ye, x3 and yz, 7, and yy, the co-ordinates 
of the four vertices, and represent the variable straight line by the 
equation ux + vy + 1=0; the two parameters wu and v will be connected 
by the relation 


(ux, + vy, +1) (Uxs + vys + 1) — Kh? (ure + vy2 +1) (ur, + V¥4 +1) =9; 


since this relation is of the second degree, it follows that the envelope 
is a curve of the second class or of the second order. The preceding 
equation is satisfied when the variable straight line coincides with one 
of the sides of the quadrilateral, since a factor in each term becomes 
zero. Hence the curve is inscribed in the quadrilateral, and one can 
assign to the ratio k a value such that the curve is tangent to any fifth 
straight line. Whence follows the general property of conic sections: 
a quadrilateral being circumscribed about a conic section, the product of 
the distances of two opposite vertices of the quadrilateral from any tan- 
gent is to the product of the distances of the other vertices from the same 
tangent in a constant ratio. 


308. TANGENTIAL EQuaTIoNn oF A Conic. —If the equation 
of a conic be 


(21) Ag? + 2 Bay + Cy? +2 Dxr+2 Ey+ F=0, 


the tangential equation of this curve is the necessary and 
sufficient condition that the straight line uz#+vy+1=0 
be tangent to it, that is ($ 126), 


(22) au? + 2 buv+er7+2du+2ev+f=0, 


an equation whose constant term f is zero if the conic be a 
parabola. 

In case the given conic (21) consists of two distinct straight 
lines, one sees that condition (22) ought to require that the 
straight line wa + vy + 1 = 0 passes through the point of inter- 
section of the two straight lines. Indeed, if a and b be the 


e 


346 PLANE GEOMETRY. BOOK III. 


co-ordinates of this point of intersection, the first member of 
the tangential equation (22) is a perfect square, the square of 


au + bv +1. 


To show this, it is sufficient to impose on this theorem the 
condition that the co-ordinates a and 6 satisfy the relations 
(Chapter XII.) 


saan 
e 


Equation (22) then becomes, on replacing the coefficients a, 
b, --- by the proportional values a’, ab, ++:, 


(au + bv + 1)’, 
which was to be proved. 
In the particular case when equation (21) represents two 
parallel straight lines, 


Cet, 
and b’—ac=— FA=0; 
equation (22) therefore becomes 

au? + 2 buv + cv? = 0, 
or (bu + cv)? = 0, 


which is still a perfect square. It is easy to show that the 


condition 
bu + cv = 0 


expresses the condition that the straight line wv + vy +1=0 
is parallel to the two straight lines represented by equation 
(21). It further follows that this condition requires that the 
straight line ua + vy + 1 =0 passes through the point of inter- 
section of the two straight lines represented by equation (21). 

When equation (21) represents two coincident straight lines, 
the first member of the tangential equation (22) is identically 
zero; one has in fact in this case 


a=b=c=d=e=f=0. 


CHAP. X. TANGENTIAL CO-ORDINATES. 347 


309. We have seen that the tangential equation of a conic 
is of the second degree inu and v. Conversely, an equation of 
the second degree between w and 2, 


23 u, v)= Au? + 2 Buv + Cv? +2 Du+2 Hv + F=9, 
) ¢ 


in which the discriminant A= ACF —.-- is different from 
zero, is the tangential equation of a conic. For, in order 
to find the envelope of the straight line 


ux +vy+1=0, 


whose coefficients satisfy the equation ¢(u, v)=0, it is suffi- 
cient to eliminate uw, v, and A from equations (17), 


a=r(Au+ B+ D), 
y=d(Bu+ Cv+ £), 
1=d\(Du+ Ev + F), 
O=A(ue +vy +1), 


where the last of equations (17) is multiplied by A. The result 
of the elimination of Av, Av, and A from these equations of the 
first degree is the condition 


DBR 5 Fate b 
Bo C2 gy =, 
| 9 eed Slated dae Ya fen 
epee Bat, 








or (24) ax? + 2bay + cy? +2de+2ey+f=90, 


the equation of a conic. It is seen that one passes from equa- 
tion (23) to (24) in the same manner as from equation (21) to 
equation (22). It is easy to verify that the tangential equa- 
tion of the conic (24) is identical with equation (28). 

It has been supposed that the discriminant A of equation 
(23) is-different from zero. If this discriminant be zero with- 
out all of its minors a, b,-+-, being zero, the function (wu, v) 
resolves itself into a product of two factors of the first degree 
in u,v. In this case the equation ¢(u, v)=0 will represent 
two points which could, moreover, be real or imaginary, and 






Pa, =, 
IBRARPy™ 
‘ ~ 
OF THE 


UNIVERSITY | 


Rts penne, 


348 PLANE GEOMETRY. BOOK III. 


the first member of equation (24) will be a perfect square, the 
square of the first member of the equation of the straight line 
joining these two points. This can be demonstrated by a 
method identical with that given above. 

Finally, if the discriminant A be zero, and also all of its 
minors, the first member of equation (23) will be a perfect 
square. It represents two coincident points. Equation (24) 
is identically zero. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 349 


CHAPTER XI* 
GENERAL PROPERTIES OF CONIC SECTIONS. 


THEOREMS OF PASCAL AND BRIANCHON. 


310. Turorem I.—/Jf three conic sections have two points 
in common, the three straight lines which join the other points 
of intersection of the curves two by two pass through a common 
point. 


Let S=0 be the equation of one of the conic sections, 
«= 0 the equation of the straight line which passes through 
the two common points; the equations of the other two conic 
sections will be of the form S —kaB=0, S— k'ya=0. The 
three straight lines which pass through the other two points 
of intersection of the curves, considered two by two, are 
B=0, y=0,kB—k'y =0; the third passes through the point 
of intersection of the first two. 


311. Turorem II.— Jf a hewagon be inscribed in a conic 
section, the points of intersection of the opposite sides are in 
a straight line. 

This theorem, which is 
- due to PAscat, is an appli- 
cation of the preceding the- 
orem. Let abcdef (Fig. 184) 
be a hexagon inscribed ina 
conic section ; the curve and 
the two pairs of straight 
lines ab and cd, af and de 
can be regarded as three 
conic sections having the — 
common points @ and d. 








350 PLANE GEOMETRY. BOOK. Iil. 


The straight line bc connects the other two points of intersec- 
tion 6 and ¢ of the curve and the two straight lines ab and cd; 
the straight line ef cormects the other two points of intersection e 
and fof the curve and the two straight lines afand de; moreover, 
the two pairs of straight lines intersect 
each other in m and p; the three straight 
lines be, ef, mp pass through the same point 
n; therefore the three points of intersec- 
tion m,n, p of the opposite sides of the 
inscribed hexagon lie in a straight line. 
This theorem is not only applicable to 
a convex hexagon, but, moreover, to a hex- 
agon formed in any manner. An inscribed 
hexagon is constructed by drawing six 
consecutive chords, in such a manner as to return finally to 
the point of departure. If the sides be numbered in the order 
in which they are constructed, the three points of intersection 
of the sides (1, 4), (2, 5), (8, 6) he on a straight line (Fig. 185). 





CorotiaAry I.—If a conic section be defined by five points 
a, b, c, d, e, the preceding theorem enables one to construct 
as many points of the curve as one may wish. Through the 
point a draw any straight line af and seek the point f where 
the straight line intersects the curve (Fig. 184); one locates 
the point of intersection m of the straight lines ab and de, 
the point of intersection p of 
the straight line cd and af; the 
straight line be intersects the 
straight line mp in a point n; 
the point jf, where the straight 
line ne intersects af, belongs to 
the curve. 

The tangent at one of these 
points can also be constructed. 
When two vertices of an inscribed 
hexagon, for example @ and f, 
coincide, the corresponding side 
af becomes a tangent to the curve at the point a; if the theo- 





Fig. 186. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 851 


rem of the inscribed hexagon be applied, on reckoning this 
tangent as a side, it follows still that three points he in a 
straight line. One locates therefore the point of intersection 
m of the sides ab and de (Fig. 186), the point of intersection 
n of the sides be and ae; the straight line cd intersects the 
straight line mn in a point p; the straight line ap will be the 
tangent at a. ' 


7 


CoroLiAry II.— A quadrilateral abcd being inscribed in a 
conic section, the points of intersection of the opposite sides, and 
the points of intersection of the tangents at the opposite vertices 
lie in a straight line. If a complete hexagon with the tangents 
at a and c be inscribed, one will have three points m, n, p in 








Fig. 187. 


a straight line (Fig. 187). If a complete hexagon with tan- 
gents at b and d be inscribed, one will have in a similar man- 
ner three points m, n, g in a straight line. Therefore the 
four points m, n, p, gq he on a straight line. 


~ CorotuaAry III.— A triangle being inscribed in a conic section, 
the points of intersection of the sides with the tangents at the 
opposite vertices are in a straight line. 


312. Remark. — We have seen that one conic section, and 
only one, can be drawn through five points a, 5, ¢, d, e, no three 
of which lie in a straight line. The elements of this curve 
can be obtained in the following manner: one begins by con- 
structing the tangents A, B, C at the three given points a, J, c. 
In every curve of the second degree the tangents at the ex- 


302 PLANE GEOMETRY. BOOK III. 


tremities of a chord intersect on the diameter conjugate to this 
chord; consequently, the straight line which joins the point 
of intersection p of the straight lines A and B to the mid- 
point g of the straight line abd is the diameter of the chords 
parallel to ab; similarly, the straight line which joins the 
point of intersection q of the straight lines B and C to the 
mid-point h of bc is the diameter of the chords parallel to be. 
Suppose that the two diameters pg and qh intersect in a 
point o. The straight line op and the straight line ok par- 
allel to ab form a system of conjugate diameters. If a’ be 
the length of the semi-diameter with the direction op, one 
has a'= Vop-og; in a similar manner the length ' of the 
semi-diameter with the direction ok may be found. It has 
been explained ($$ 174 and 195) how to determine the axes, 
in case a system of conjugate diameters a’ and 6’ are known. 

If the two diameters be parallel, the curve is a parabola. 
In this case, one draws the diameters which pass through 
«a and b, then the straight lines forming with the tangents 
angles which are equal to those formed by the diameters with 
the tangents; these two straight lines intersect at the focus 
of the parabola. On dropping from the focus perpendiculars 
to the tangents A and B, and prolonging each of the perpen- 
diculars a length equal to itself, two points of the directrix 
are determined. 

In case three points and the tangents at two of these points 
are given, the tangent at the third point is determined by 
means of the property of the inscribed triangle; after this one 
proceeds as above. The construction relative to the parabola 
can evidently be used in case two tangents to the curve and 
their points of contact be known. 





313. Suppose, finally, it is desired to find the elements of 
a parabola determined by four points a, 6, ¢, d. If the two 
straight lines ab, cd be chosen as axes of co-ordinates, the 
equations of the parabolas passing through the given points 
are (§ 276) 

eT i sare 8 
ab” ~/abed ca 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 353 


Since the angular coefficients of the axes of the parabolas are 
+4 Ee it follows that these axes are parallel to the diagonals 
a 


of a parallelogram constructed on the axes of co-ordinates and 
of which the sides will have lengths which are mean propor- 
tionals between a and b,c and d. Knowing the direction of the 
axis, the theorem concerning the inscribed pentagon will give, 
on supposing that the point e be removed indefinitely, that is, 
that the straight lines ae and de, for example, become parallel 
to the axis (Fig. 186), the tangent at one of the points. In 
case two tangents have been determined, the problem will be 
reduced to the preceding case. oe 


314. THreorem IIT.— Jf a hexagon be inscribed in a conic 
section, the three straight lines which join the opposite vertices 
pass through the same point. 


This theorem, discovered by BriaAncuon, may be derived from 
the preceding by the method of reciprocal polars. Let abcdef 
(Fig. 188) be a hexagon circum- 
scribed about a conic section; 
the inscribed hexagon, which 
has as vertices the points of 
contact, is the corollative figure 
of the circumscribed hexagon, 
with respect to the given conic 
section; because the vertices 
a, b, c,--- of the circumscribed 
hexagon are the poles of the 
sides A’, B’, C’,--- of the in- 
scribed hexagon. The diagonal 
ad of the circumscribed hexagon 
is the polar of the point of in- 
tersection m' of the opposite 
sides A' and D' of the inscribed 
hexagon; similarly, the diago- 
nal be is the polar of the point 
of intersection n' of the sides 
B' and E', and the diagonal cf the polar of the point of 


Z 





I 
i Fig. 188. 


ne: PLANE GEOMETRY. BOOK III. 


intersection p' of the sides C' and F". Since the three points 
m', n', p' are in a straight line, the three straight lines ad, be, 
cf pass through the same point 0, the pole of this straight 
line. 

We make at this point a remark analogous to that which has 
been made with respect to the theorem of Pascan. It is not 
necessary that the circumscribed hexagon be convex, it is suffi- 
cient that it be closed. Suppose that six tangents be drawn to 
a conic section; in order to construct the hexagon, beginning at 
the point of intersection of two tangents, one proceeds along 
one of them to the intersection of the next tangent; then 
along this second tangent, in either direction, to the intersec- 
tion of a third tangent, and so on, in a similar manner, till the 
point of departure is reached, after having traveled along 
the tangents in a continuous manner. The broken line thus 
formed is a circumscribed hexagon. If the vertices be num- 
bered in the order in which they are constructed, the three 








Fig. 189. 


diagonals which connect the vertices (1, 4), (2, 5), (8, 6) pass 
through the same point (Fig. 189). 

CoroLuAry. —If a conic section be defined by five tangents, 
one can, by aid of the preceding theorem, construct as many 
tangents as one wishes. Let the five tangents be ab, be, cd, 
de, ef (Fig. 188); determine the second tangent which passes 
through a point a taken arbitrarily on one of the given tan- 
gents; take the point of intersection o of the diagonals ad and 
be, and draw the straight line co and join the point a to the 
point f, where the straight line co intersects the tangent ef. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 350 


The point of contact of each of the tangents can also be 
determined if two sides of the circumscribed hexagon, for 
example the sides ab and be, coincide; the intermediary vertex 
b becomes the point of contact; in order to find this point of 
contact, one connects the vertex e with the point of intersec- 
tion o of the diagonals ad and cf (Fig. 190). 

When the points of contact of the three tangents have been 
determined, one obtains the elements 
of the curve by the method which we 
have described in § 312. 

The center could also be immedi- / 
ately obtained by means of the 
theorem demonstrated in § 300. 

The following corollaries may be 
deduced from the theorem of BriAn- 
cHon: If a quadrilateral be circum- Fig. 190. 
scribed about a conic section, the two diagonals and the two 
straight lines which join the points of contact of the opposite sides 
pass through the same point. 

If a triangle be circumscribed about a conic section, the straight 
lines which join the vertices to the points of contact of the opposite 
sides pass through the same point. It is sufficient to complete 
the circumscribed hexagon in the first case with the points of 
contact of the two opposite sides, in the second case with the 
three points of contact. 





HOMOGRAPHIC SYSTEMS. 


315. In case we are given on two given straight lines two 
systems of points which have a one-to-one correspondence, of 
the kind that if a and aw! represent the distances (affected with 
the proper signs) of two corresponding points from two fixed 
points taken on the straight lines, one has the relation 


(1) Axa' + Ba+ Cz'+ D=90; 


these two systems of points are said to be homographic. 
This equation involves three arbitrary parameters; to three 
points taken at will on the first straight line there can be made 


356 PLANE GEOMETRY. BOOK III. 


to correspond three points taken at will on the second; this 
mode of homographic division is therefore perfectly deter- 
mined. 

When the point m' of the second straight line is removed 
to infinity, the homologous point m of the first approaches a 
limiting position 7 given by the formula « =— <. Simuarly, 
in case the point m of the first straight line is removed to 
infinity, the point m' of the second approaches a limiting 


position j' given by the formula a’ = _" If one lay off the 


distances on these two straight lines, beginning with the 
points j' and 7, the relation is simplified and becomes 


(2) Axxz'+ D=0. 


A pencil of straight lines which passes through the same 
point o (Fig. ea determines on any two secants two systems 
of homographic points; be- 
cause, on calling a, and y, the 
co-ordinates of the point o 
with respect to the two secants, 
a and £8 the abscissa and the 

ae cae, ordinate of the points m and 

nee m', when one of the straight 

lines of the pencil intersects the two secants, the eo 
a and B will be connected by the relation 


vc 








1 OY 
Fann ae 
Conversely, when one has two systems of homographic 
points on two straight lines, the straight lines can be so 
placed that one of the systems is the perspective of the other; 
it is sufficient to place one of the straight lines so that two 
peo Ceous points a and a! coincide; the straight lines 0b’, 
cc’, which join two pairs of homologous points, intersect in a 
point 0; the straight line om, which joins the point o to any 
point m of the first straight line, w ill pass through the homol- 
ogous point m! of the second. The straight lines of and oj’, 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 307 


parallel to the two straight lines, give the points ¢ and j’.. Two 
systems of points homographic to a third are homographic to 
each other; because the elimination of a from the two equa- 
tions 


(Av' + B)x + (Cz' + D) =0, 
(A,x, + B,) e+ (C\2, + D)) = 9, 


gives an equation in #' and a similar in form. 


316. Consider the two pencils of straight lines which are 
obtained by-connecting two fixed points o and o' with two 
systems of homographic points; these two pencils determine 
on any secant systems of homographic points; accordingly the 
two pencils of straight lines are said to be homographie. 

Imagine that through a fixed point situated on the a-axis, 
and having the abscissa —1, one draws lines parallel to the 
straight lines of the two pencils; these parallels determine on 
the y-axis two systems of homographic points; the ordinate of 
each of these points being equal to the angular coefficient 
of the corresponding straight line, one infers that the angular 
coefficients m and m’ of the homologous straight lines are 
connected by the relation 


(3) Amm'+ Bm + Om'+ D=0. 


Conversely, if in two pencils the angular coefficients of two 
homologous straight lines satisfy a relation of this form, the 
two pencils are homographic. Such, for example, are the two 
pencils which one obtains by drawing through two fixed points 
o and o’ lines parallel to two conjugate diameters of a conic. 

Two pencils homographice to a third are homographic to 
each other. 

Two systems of homographic points are transformed in the 
reciprocal polar figure into two pencils of homographic straight 
lines. Let P and Q be the two given straight lines, p’ and q’ 
their poles with respect to a curve of the second degree whose 
center is 0; to a point a of the first straight line corresponds 
a straight line A’ passing through its pole p'; to a point b of 
the second straight line a straight line B' passing through its 


358 PLANE GEOMETRY. BOOK II. 


pole g'.. The straight lines oa, ob form two homographie 
pencils; the pencil of straight lines A’ being homographic with 
that of the straight lines oa and the pencil of straight lines 
B' with that of the straight line od, it follows that the two 
pencils A' and B’ are homographie. 


Remark. — In the two relations (1) and (8) we have sup- 
posed that the coefficients A, B, C, D do not satisfy the 
condition 

AD— CB=0. 

If this condition were fulfilled, the first member of relation 
(1), for example, would resolve into two linear factors, the 
one in # the other in 2’, and this relation (1) would take the 
form . 

A(a — a) (x! — «') = 0. 

To a value assigned to w' would always correspond the value 

2 = a, and for #'= «', x would be indeterminate. 


317. In case two systems of homographic points lie on the 
same straight line, there exist two double points on this line; 
that is, two points such that either of them, considered as 
belonging to one of the systems, coincides with the other, its 
homologous point in the other system. In fact, if one lay off 
the distances on the straight line, beginning with the same 
point, and put «#’= 2, one has, owing to relation (1), an equa- 
tion of the second degree, e 
(4) Av’? +(B+C)x+D=0, 
each of whose roots gives a double point. The two double 
points are real or imaginary. 

Suppose that there have been constructed, as already has 
been explained, the two points ¢ and j' homologous with respect 
to infinity; if one lay off the distances beginning with the 
point ¢, mid-point of <j’, equation (1) becomes 


(5) Agx'+ Bia —2')\+ D=0. 
Equation (4), which gives the double points, reduces to 


(6) Ax? + D=0. 


CHAP. XI PROPERTIES OF CONIC SECTIONS. 359 


Call c! the point of the second system homologous to the point 
c of the first; equation (5) ought to be satisfied by ¢=0 and 
z'=cc'; one has 


: oa 
Be 
moreover, qj =— a 5 
A 
whence it follows a= —cj' x ec’, 
and equation (6) becomes 
(7) = cj' x ce’. 


The double points are real in case the two lengths ¢j' and cc' 
are measured in the same direc- 

tion; in order to construct them, oe 
a circle is constructed on c'j’ as a ; : 
diameter (Fig. 192); a tangent eer ae Beer TT e 
is drawn from the point ¢ to this 
circle; by revolving the tangent 
to the straight line, the two double points e and f are deter- 
mined, which are situated at equal distances from c. 


= 
-- - 
o~ 





Fig. 192. 


318. Two homographie pencils having a common vertex have 
in like manner two double straight lines, real or imaginary ; 
their equations may be obtained by joining the vertex to the 
two double points of the homographie division determined by 
the pencil and any secant. 

In case a constant angle is made to revolve about its vertex, 
the two sides form two homographic pencils with this point as 
&% common vertex, the various positions of one of the sides 
constituting the first pencil, those of the second side the other 
pencil. Because if the first pencil revolve through a constant 
angle about the vertex, it coincides with the second. The rela- 
tion between the angular coefficient of the homologous straight 
lines in rectangular co-ordinates is mm!'+1-+ ¢(m—m')=0; 
the double straight lines are imaginary and have the equation 
2? +42=0; they are the asymptotes of the circle & + Y=”. 


360 PLANE GEOMETRY. BOOK IIl. 


These two pencils determine on any straight line two Sys- 
tems of homographic points, whose double points are imaginary. 
Conversely, in case the double points of the two systems of 
homographie points on the same straight line are imaginary, 
these two systems of points can be obtained by the rotations 
of a constant angle about its vertex. The mode of division is 
defined by the three pairs of 











a’ 0 a 
points (¢, ¢’), (i, 0), (a, j'), the 
point ¢ being the mid-point of ¢j'; 
____ co, the perpendicular to the straight 
: : ag a ’ line, intersects the circle described 


on c/j' as diameter in a point 0; 
the angle c'oc, on revolving about the point 0, will give the 
homographic division desired. : 


319. InvoLtution. — Consider two systems of homographic 
points on the same straight line, and suppose that two homo- 
graphic points @ and a’ be reciprocal, that is, that if to the 
point a of the first system there corresponds the point a! in 
the second, reciprocally to the point a’ considered as belonging 
to the first system there corresponds the point « in the second. 
It follows that equation (1) will be satisfied when the par- 
ticular values of # and x’ which belong to these two points are 
permuted, which requires that B= C; but in this case all of 
the homologous points are reciprocal two by two, and the 
points are said to be in involution. - 

The equation 


(7) Axx'+ Bix + 2')+ D=0 

containing but two arbitrary parameters, two pairs of conjugate 
points (a, a’), (b, 6’) are sufficient to define the involution. 
The two points ¢ and j’ coincide, and if the distances beginning 
with this point 7 be laid off, equation (7) becomes 

(8) Agz'+ D=0; 

this point is called the center of involution. There are two 


double points e and f, real or imaginary, given by the equation 


Ax? + D = 0. 


- 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 361 


Equation (8) becomes, therefore, 2a’ = ie’; it follows that the 
two double points e and f are harmonic conjugates with respect 
to any two conjugate points. 

The circles drawn through the two points p and g determine 
on a straight line an involution 
(Fig. 194). Let i be the point 
in which the straight line pq in- 








tersects the given straight line; if _¢ ae | ite 
x and a! be the distances of this y Pe 
point from the two points of inter- 

: Fia. 194. 


section of the secant and of one of 
the circumferences, one has xx'!= ip-ig; the point ¢ is there- 
fore the center of involution. The double points are real or 
imaginary, according as the point 7 is situated without the 
points p and q or between these two points. In the first case, 
one obtains the double points on drawing from the point 7 a 
tangent to one of the circles and revolving this tangent. 

In case the involution on a straight line is defined by means 
of two pairs of conjugate points (a, a’), (6, b'), any two con- 
jugate points can be easily constructed; construct a circle 
through the two points a and a’, construct also a second circle 
through the two points } and b' and an arbitrary point p of the 
first; these two circles intersect in a second point q; the circle 
which passes through the two points p and q and a point m of 

the straight line will determine the conjugate point m!. 

Let us consider in like manner two homographic pencils 
with a common vertex and such that two homologous straight 
lines are reciprocal; these straight lines determine on any 
secant the points of involution; all homologous straight lines 
are therefore reciprocal two by two and the straight lines are 
said to be in involution. Two double straight lines may be 
real or imaginary. 

We have mentioned (§ 318) that, if a constant angle revolve 
about its vertex, its sides form two homographic pencils. When 
the angle is a right angle the pencils will be in involution; the 
double straight lines, as has been remarked, are the asymptotes 
of a circle. 

Conversely, in case the double points of an involution on a 


362 PLANE GEOMETRY. BOOK IIt. 


straight line are imaginary, the pairs of conjugate points can 
be found by the rotation of a right angle about its vertex. The 
involution is defined by the two 
pairs of conjugate points (7, 0), 
(a, a'); describe a circle on aa’ 
as a diameter (Fig. 195); erect 
a perpendicular to the straight 
line at the point 7 which inter- 
sects this circle in two points p 
and q; a circle passing through 
the two points p and q will determine two conjugate points 
m and m', and the angle mpm! is a right angle. 

The conjugate diameters of the conic are in involution. The 
double straight lines are real in case of the hyperbola, imagi- 
nary in case of the ellipse. 


‘7 








Fig. 195. 


320. Turorem I.—Jf two homographic pencils be given, the 
locus of the point of intersection of two homologous straight lines 
is a conic passing through the vertices of the two pencils. 


Determine as many of the points of the locus as are situated 
on any straight line D; the two homographie pencils o and o' 
(Fig. 196) determine on this straight line two systems of 

kau | homographic points (a, @'), (8, B'), 
(y, y'),:+23 two homologous straight 
lines oe, o'e, which intersect on the 
p straight line D, determine a double 
point e; since there cannot be on the 
straight line D more than two double 
points e and f, it follows that this 
straight line intersects the locus in 
but two points, real or imaginary; hence the locus is of the 
second order. 

To the straight line o'o of the second pencil corresponds a 
certain straight line op of the first; the point of intersection 
falls in 0, and the straight line op is tangent to the curve at 
this point. Similarly, the curve passes through the point o’ and 
is tangent at this point to the straight line o'g' of the second 
pencil, the homologous line of the straight line oo! of the first. 






Fig. 196. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 363 


CoroLLaRY. — This enables us to find the points in which 
a given straight line D intersects a conic defined by five 
points, 0, o', a, b, c; if the two points o and o' be joined to 
the other three, one has three pairs of straight lines (0a, o'a), 
(ob, 0'b), (oc, o'c), which determine the two homographic 
pencils 6 and 0’; the locus of the point of intersection of the 
homologous straight lines is the conic passing through the 
five given points; the three pairs of points: (a, «'), (B, B’), 
(y, y') define the homographic division on the straight line 
D; the two double points e and f may be found by the method 
described in § 317. 

If the straight line pass through one of the given points, 
for example o, it is sufficient to construct the homologous 
straight line in the second pencil. Similarly, as we have 
already said, the tangent at o may be found by drawing the 
straight line op of the first pencil homologous to the straight 
line o'o of the second. Thus may be found as many points 
and tangents of the conic sought as one wishes. 


REMARK. — When the straight line 00', which passes through 
the vertices, corresponds to itself in the two pencils, it evl- 
dently constitutes a part of the locus which is then composed 
of two straight lines; in this case, the locus of the point of 
intersection of the homologous straight line is, strictly speak- 
ing, a straight line. 


321. Turorem II.— Jf two systems of homographic points 
on two fixed straight lines A and A' be given, the straight line 
aa', which joins any two homologous points, envelops a conic 
which is tangent to the two fixed straight lines. 


Determine all the tangents to the envelope which pass 
through an arbitrary point p of the plane (Fig. 197); the 
straight lines pa, pa', which join the point p to two homol- 
ogous points, form about the point p two homographic pencils; 
in case the variable straight line aa', in one of its positions 
mm', passes through the point p, it becomes a double straight 
line of the two pencils; since there can exist but two double 
straight lines pm, pn, it follows that through the point p there 


364 PLANE GEOMETRY. BOOK III. 


can be drawn to the envelope curve but two tangents, real or 
imaginary; this curve is therefore of the second class, and 
consequently of the second order. 

At the point of intersection o of the two fixed straight lines 
A and A', considered as belonging to the second straight line, 





Fig. 197. 4 


there corresponds a point h on the first; the variable straight 
line coincides with oh, and the curve is tangent to the straight 
line a at the point h. Similarly, the curve is tangent to-the 
straight line A' at the point g’ of this straight line homologous 
to the point o of the straight line A. 


Corotiary.— This theorem enables us to draw through a 
given point p tangents to a conic defined by five tangents; 
if one join to the point p the points where the two tangents 
A and A’ are intersected by the other three, B, C, D, one 
obtains three pairs of straight lines determining two homo- 
graphic pencils whose double straight lines are the tangents 
required. 

If the point p be situated on one of the given tangents, A 
for example, the points where the tangents A and A’ are 
intersected by the other three, B, C, D, determine on these 
first two tangents two systems of homographic points; one 
seeks on the straight line A’ a point p’ which is homologous 
to the point p on A; the straight line pp! will be tangent to 
the conic. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 369 


The point of contact of the tangent A is, as has been men- 
tioned; the point of this straight line which is homologous to 
_the point o of A’. 7 


RemMArkK.—In case the point of intersection o of the two 
fixed straight lines corresponds to itself on the two straight 
lines, in the reciprocal polar figure the straight line of the 
vertices will correspond to itself in the two pencils; the locus 
becomes in this case a straight line, the envelope reduces to 
a point. Therefore every straight line, such as aa', passes 
through the same point. 


322. The two preceding theorems give rise to a large num- 
ber of remarkable properties. We shall now call attention to 
some of them. 

For example, if two constant angles revolve about their 
vertices in such a way that the point of intersection of two 
sides describes a fixed straight line, the other two sides will 
form two homographic pencils, and, consequently, the locus 
of their point of intersection will be a conic passing through 
the two fixed vertices. 

Similarly, if, on two fixed straight lines, one begin with the 
points where they are intersected by a variable secant drawn 
through a fixed point, and lay off in a definite manner two 
constant lengths, it is evident that the extremities of these 
lengths will form two systems of homographic points, and 
consequently that the straight line which connects them will 
envelop a conic tangent to two fixed straight lines. 

Let us consider a triangle maa', of which the three sides 
revolve about the three fixed points 0, 0', p (Fig. 198), whilst 
the two vertices a and a’ slide . 
along the two fixed straight lines 
A and A’; the pencils o and p are 
homographic; similarly, p and o'. 
Therefore the pencils o and o! are 
homographic, and the point m, the 
third vertex of the triangle, de- 
scribes a conic passing through Fig. 198. 
the two points o and o'. It is easy to see that the point 








366 PLANE GEOMETRY. BOOK III. 


of intersection c of the straight lines A and A' and the two 
points d and e, where these straight lines are intersected by 
the straight lines po' and po, belong to the locus; thus the 
conic is defined by five points. 

When the three fixed points 0, o', p lie in a straight line, 
the straight line 00' corresponds to itself in the two pencils, 
and the locus of the vertex m is a straight line; this problem 
has been discussed in § 105. 

Similarly, let us consider a variable triangle aba' (Fig. 199), 
whose three vertices slide on three fixed straight lines A, A’, 
B, while the two sides ba, ba! re- 
volve about the two fixed points 
o and o'; the pencil o determines 
on the straight lines A and B two 
systems of homographic points a 
and 6; similarly, the pencil o' deter- 
mines two systems of homographic 
points 6 and a’. Therefore the two 
systems a and a’ are homographie, 
and the third side aa’ of the triangle envelops a conic tangent 
to the two straight lines A and A’. It can be easily verified 
that the straight lines o'e and od, which join the points o and o' 
to the points where the straight line B intersects the straight 
lines A and A‘, touch the conic; thus the conic will be defined 
by five tangents. 

If the three straight lines A, A', B pass through the same 
point, the point of intersection of the straight lines A 
and A’ corresponds to itself, and the envelope reduces to a 
point. Therefore the straight line aa’ passes through a fixed 
point. 

This mode of demonstration is applicable to polygons of any 
number of sides. Thus, if the » sides of a polygon revolve 
about fixed points, and n — 1 vertices describe straight lines, 
the last vertex describes a conic. In case n vertices of a 
polygon describe straight lines, and » —1 sides revolve about 
fixed points, the last side envelops a conic. 

Theorems I. and II. make it possible, as we have seen, to 
construct a conic defined by five points or five tangents; but 








Fig. 199. 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 367 


the theorems of Pascal and Brianchon furnish more simple 
constructions. 


324. Turorem III. — In case, in the general equation of a 
pencil of conics subject to four conditions, the arbitrary parameter 
appears in the first degree, these conics determine on any straight 
line points in involution. 

If the straight line be chosen as the a-axis, and if y be made 
equal to zero in the given equation, one obtains an equation of 
the form 

(A+kA' a? +(B+ kB')a+(C+kC')=0, 


k being the arbitrary parameter. On calling x and a? the two 
roots, one has 
ee ints ee tee 
—B—kB! C+k0 A+kA"’ 
whence it follows 
A'a+a\+B'  —A've'+C! 
AR BA AC ee 














Q.E.D. 


325. Turorem IV.— Conics which pass through fowr given 
points determine on any straight line points in involution. 

We have seen (§ 277) that the equation of conics, which 
pass through four given points a, 6, c, d, involves one arbi- 
trary parameter in the first degree; according to the preceding 








theorem these conics determine on any straight line D points 
in involution (Fig. 200). The pairs of straight lines (ac, bd), 
(ab, cd) determine two pairs of conjugate points («, @'), (8, B'), 
which define the involution. 


368 PLANE GEOMETRY. BOOK IIt. 


CoroLLARy.—The double points of involution are the points 
of contact of the conics which pass through the four given 
points, and are tangent to the straight line D; since there 
are two double points, it follows that there are two conics, 
real or imaginary, which pass through four given points and are 
tangent to a given straight line. These points are determined 
by the construction described in § 319, and then each of the 
conics will be defined by five points. 


326. THErorem V.— The tangents drawn from a fixed point 
to conics tangent to four given straight lines are in involution. 


This theorem may be deduced from the preceding, which 
is due to Drsarcurs, by the method of reciprocal polars. 
To the conics tangent to four given straight lines there cor- 
respond, in the reciprocal polar figure, conics which pass 
through four given points; to the two tangents drawn from 
a fixed point p to one of the first system of conics corre- 
spond the two points of intersection of the straight line P, 
the polar of the point p, with one of the second system of 
conics; these points of intersection on the straight line P' 
being in involution, the pencils of tangents emanating from 
the point p’ are also in involution. 

The four given straight lines (Fig. 201) form a quadrilateral ; 
the diagonal aa can be regarded as the limit of an ellipse 
tangent to the four straight lines, and of 
which the minor axis becomes zero; the 
tangents drawn from the point p to this 
ellipse, reduced to its major axis, are 
pa and pa’. A similar discussion apples 
to the diagonal bb’. One has therefore two 
pairs of conjugate straight lines (pa, pa’), 
(pb, pb') which define the involution. 


Coro.LuarRy. — If the conic pass through 
the point p, the two tangents pm, pm' co- 
incide and form a double straight line; 
since there are two double straight lines 

Fig. 201. in the involution, it follows that there are 
two conics, real or imaginary, which touch four given straight 








CHAP. XI. PROPERTIES OF CONIC SECTIONS. 369 


lines and pass through a given point. On drawing a secant 
across the pencil, and determining the double points on the 
secant, one will obtain the double straight lines, and each of 
the two conics will be defined by five tangents. 


327, THrorem VI.— The conics tangent to two given straight 
lines at two given points determine on any secant an involution 
of which one of the double points is situated on the chord of con- 
tact. 

This theorem is a particular case of Theorem IV. Suppose 
that the points a and c coincide, also that the points b and d 
(Fig. 200); the two straight lines ac and bd will be tangent 
at a and b; the two straight lines ab and ed coinciding, the 
two conjugate points B and f' coincide with one of the double 
points of the involution, to which belong the pairs of points 
(m, m'), (a, @'). 

CoroLuARy. — This theorem enables one to construct a conic 
which passes through three given points and touches two given 
straight lines A and A! (Fig. 202). 






é, 


{% 





a b a Qa’ 
SOR Igete oN. 


Select on the secant ab the two double points e and f of the 
involution defined by the two pairs of points (a, 6), (a, @’). 
Select in a similar manner on the secant ac the two double 
points e, and f, of the involution defined by the two pairs 
of points (a, ¢), (a, @')). 

The chord of contact, passing through one of the two points 
e and jf, and through one of the two points e, and fi, will 
coincide with one of the four straight lines which are found 

2A 


870 PLANE GEOMETRY. BOOK III. 


by connecting these points two by two in all possible ways. 
Any of these four straight lines, for example ee,, will give 
a solution of the problem: the straight line ee, intersects the 
two given straight lines A and A’ in two points m and m'; 
one conic can be drawn through the point a and tangent to 
the straight lines A and A’ at the points m and m! (§ 280); 
this conic intersects the secant aa!’ in a second point conjugate 
to the point a in the involution defined by the double point 
e and the pair of points (a, «’), will pass through the point 0; 
it can be shown in a similar manner that the conic passes 
through the point c. Hence there are four conics, real or 
imaginary, which pass through three given points and touch 
two given straight lines. 


328. Turorem VII.— The tangents drawn from a fixed point 
to the various conics which touch two given straight lines in two 
given points, form an involution of which one of the double 
straight lines passes through the point of intersection of the two 
given straight lines. 

This theorem is a particular case of Theorem V. Suppose 
that the two tangents ab and ab! coincide (Fig. 201), also that 
a'b and a'b': the points a and a’ become the points of contact 
of the tangents ab and a'b'; the two points b and b' coincide ; 
the two straight lines pb and ph' coincide with one of the 
double straight lines of the involution, to which belong the 
two pairs of straight lines (pm, pm’), (pa, pa'). 


CoroLtLaRy. — The preceding theorem enables one to con- 
struct a conic passing through two given points a and b, and 
touching three given straight lines mn, pm, and pn (Fig. 203). 
The point of intersection o of the tangents at a and b is situ- 
ated on one of the two double straight lines of the involution 
defined by the two pairs of straight lines (pa, pb), (pm, pn), 
and on one of the two double straight lines of the involution 
defined by the two pairs of straight lines (ma, mb), (mn, mp), 
will coincide with one of the four points of intersection of 
these double straight lines taken two by two. Any one of 
these four points, for example the point 9, will give a solu- 


CHAP. XI. PROPERTIES OF CONIC SECTIONS. 871 


tion of the problem; one conic tangent to the two straight 
lines oa and ob at the points a and b and to the straight line 
pm can be determined; the second 
tangent which can be drawn from 
the point p to this conic will be the 
conjugate of the straight line pm 
in the involution defined by the 
double straight lite po, and the 
pair of straight lines (pa, pb) will 
coincide with pn; in a similar man- 
ner it can be demonstrated that the 
straight line mn is tangent to the 
conic. Hence there are four conics, 
real or imaginary, passing through 
two given points and touching three given straight lines. 





Fig. 203, 


329. Remark. — It has been stated (§ 283) that a focus can 
be regarded as the point of intersection of two tangents whose 
angular coefficients are + 7, that is, tangents that are parallel 
to the asymptotes of a circle; to be given a focus is therefore 
equivalent to being given two tangents to the conic. Hence, 
of the conics which have a given focus in common, there is 
one tangent to three given straight lines ($ 262), two tangent 
to two given straight lines and passing through a given point, 
four (of which two are real and two imaginary) tangents to 
one given straight line and passing through two given points, 
and, finally, four passing through three given points (§ 260). 

We have learned to construct a conic which satisfies five 
simple conditions, points of the curve or tangents; four condi- 
tions are sufficient for the determination of a parabola, and the 
discussion can be reduced to one of the preceding by a transfor- 
mation with the assistance of the method of reciprocal polars. 
We know, in fact ($ 299), that if the center o of the curve of 
reference be situated on a conic, the polar reciprocal curve is a 
parabola, and that, conversely, the reciprocal polar curve of a 
parabola is a conic passing through the center o of the curve of 
reference. In the transformation, the condition that the curve 
sought is a parabola, is therefore replaced by the point 9, the 


Siz PLANE GEOMETRY. BOOK III. 


points by the straight lines, the straight lines by the points. 
The construction of a parabola tangent to four given straight 
lines is thus reduced to the construction of a conic passing 
through five given points; there is one solution, and one only. 
Similarly, there are two parabolas passing through four given 
points, or passing through one point and tangent to three given 
straight lines; fowr parabolas passing through three points 
and tangent to one straight line, or passing through two points 
and tangent to two given straight lines. On drawing the tan- 
gent to the reciprocal polar curve at o and the conjugate 
diameter in the curve of reference, one will have the direction 
of the diameters of the parabola, which makes it possible to at 
once apply the preceding given theorems. 

M. Chasles has conceived an ingenious method for studying 
the properties of a system of conics which satisfy four given 
conditions, and he showed that these properties depend upon 
two integral numbers which he called the characteristics of the 
system; these represent the number of conics of the system 
which pass through a given point or which touch a given 
straight line. For example, the two characteristics of a sys- 
tem of conics which pass through four given points are 1 
and 2; those of the system of conics which touch four given 
straight lines are 2 and 1; those of the system of conies which 
pass through three-given points, and which touch one straight 
line, are 2 and 4; those of the system of conics which pass 
through one point, and which touch three straight lines, are 
4 and 2; finally, those of the system of conics which pass 
through two points, and which touch two straight lines, are 
4 and 4. 


HOMOGENEOUS CO-ORDINATES. 


330. When an algebraic curve defined by its equation 
F(a, y)=0 is investigated, it is an advantage to consider the 
homogeneous integral function obtained by replacing the co- 
ordinates « and y in F(a, y) by oe and multiplying the 

Z 2 


result by a suitable power of z. One has an illustration of 
this method when one seeks the equation of the tangent at a 


CHAP. XI. HOMOGENEOUS CO-ORDINATES. 373 


point of the curve (§ 291), or the co-ordinates of the point of 
contact of the tangents drawn from any point of the plane. 
Three numbers 2, y, z are called the homogeneous co-ordi- 


nates of a point, if the ratios - z be respectively equal to the 


abscissa and ordinate of this point. Thus a point whose ab- 
scissa is 2 and ordinate 4 has as homogeneous co-ordinates 9, 2, 
12 or 9n, 2n, 12”, n being any number different from zero. 
Let, moreover, f(a, y, z)=90 be the equation of an algebraic 
curve rendered homogeneous by the method which we have 
described; this equation is called the equation of the curve 
in homogeneous co-ordinates. 


Points AT INFINITY.—StTrRaicHT Line at INFINITY. — 
According to the preceding definition for any point of the 
plane, the third of the homogeneous co-ordinates z is never zero. 
One considers, nevertheless, the system of values of a, y, 2 
in which the third variable z is zero, and it is said that such 
a system (2, y,, 0) corresponds to a point at infinity, and that 
this point is on a curve whose equation in homogeneous co-ordi- 
nates is f(a, y, 2)= 0, if one have f(a, th, 0)= 0. 

In particular, to say that the point at infinity (a, 7, 0) 1s on 
the straight line y = ax + bz, is to say that one has the condi- 
tion y, = aa, It is to be remarked that, in order to justify 
this representation of the point at infinity, one is led to the 
consideration of such systems of values of a, y, z as when the 
point is supposed to be moved continuously along a given 
straight line toward infinity. Thus if the homogeneous co- 
ordinates of the point be 


@ = 2 — ANy Y= IY. —AYy P=1—dD, 
and, consequently, its Cartesian co-ordinates 


@ — AM, I — AYo 
1-rA  1-A 





the point is on the straight line whose homogeneous equation is 
ee) Sa 
% y 1/=9, 
® Yo I 


374 PLANE GEOMETRY. BOOK III. 


iat 


value 1, the point approaches infinity, and for the value A = 1, 


the corresponding values of a, y, z are 


whose angular coefficient is a= If X approach the 


XY — Loy Yi — Ya 0, 


which satisfy the condition y — ax = 0. 

Since the homogeneous equation of the first degree in a, y, z 
represents, in general, a straight line, and that the co-ordinates 
of all the points at infinity satisfy the homogeneous equation 
of the first degree z=0, it is said that all the points at in- 
finity are situated on a straight line (the straight line at 
infinity) whose equation is z=0. Accordingly the two parallel 


straight lines 
y=ax+ bz, y=ax+b'z 


are said to have a common point at infinity, or intersect on 
the straight line at infinity; for the equations of these two 
straight lines are satisfied by the same system of values of 
UY; @: | 

ae 
in which z 1s zero. Similarly it may be said that the two 
curves 


J (2%, y, 2) =0, (@, y, 2) =9 


have a common point (a, 7, 0) at infinity if one have 
SF (@y Yy 0) = 0, (ty, H, 0) = 0. : 

As in § 270, an imaginary point is considered as a system of 
imaginary values of a, y, z, with the condition that one cannot 
make a, y, z real on dividing them by the same imaginary 
quantity. An imaginary point at infinity will be an imaginary 
point of which the co-ordinate z is zero. 

Exampier.— The general equation of a circle in rectangular 
Cartesian co-ordinates x and y is 


“e+y+tax+by+c=), 
therefore in homogeneous co-ordinates, 


a + y? + (ax + by + cz)z = 09. 


CHAP. XI. HOMOGENEOUS CO-ORDINATES. S75 


This equation is satisfied by the two systems 
gon f, y=V—1, z= 0; eat 1, y=—-v-1, z=, 

whatever values a, 0, c may have. It can therefore be said 
that any circle whatever passes through the points at infinity 
whose homogeneous co-ordinates are (1, V—1, 0), A, —Vv 1, 
0). These twg points are called the circular points at infinity. 

Conversely, every curve of the second order which passes 
through the circular points at infinity is a circle. Because 
if the general equation of the second degree 

Ag? + 2 Bry + Cy? + 2 Dxz + 2 Eyz + F2 =0 
be satisfied by the co-ordinates of these two points, one has 
A+2BV—1—C=0, A—2BV—1-—C=0, 
whence, by adding and subtracting, 
Bite OF 3a 


which shows that the conic is a circle. 


FoRMULAS OF TRANSFORMATION. 


Suppose that one makes a change of co-ordinates, and that 
the formulas of transformation for the Cartesian co-ordinates 


are 
7 Xl=at+mxX-4+ny, 
©) Yi=b-+ pX + q¥, 
one will take, as homogeneous co-ordinates, the,formulas 

v= az+ mar+ ny, 
(2) y! = be + px t ay; 

g'== 2%, 
If the point (a, y, z) be at a finite distance (z different from 
zero), these formulas are in fact identical with the formulas 
(1) according to the definition of homogeneous co-ordinates. 
If, on the contrary, z be zero, that is, if the point (a, y, z) be 
at infinity, it follows by definition that one will regard the 
values 2’, y', 2', given by formulas (2), as the new co-ordinates 
of the same point; it is to be noticed that one still has 2’ = 0. 


376 PLANE GEOMETRY. BOOK III. 


APPLICATIONS. 


Seek the equation in homogeneous co-ordinates of the 
straight line which joins the two points M, and M;, which 
have as co-ordinates (a, 4, 2), (a2 Yo, z,). This equation is 


Cy 
(3) m% Yy %|=O0: 
vq Yo 2 


in fact, the equation which has just been written represents a 
straight line, and this straight line passes through the two 
points, because the equation is evidently satisfied by the 
co-ordinates of the two points. One can express, as follows, 
the co-ordinates of any point of the straight line as a function 
of a parameter. The determinant (3) being zero, there exists 
a linear homogeneous relation between the elements of the 
three columns: 


Ax + Bx, +Cx,=0, Ay+ By, +Cy,=0, Az+ Bz, + Cz, = 0. 


The coefficient A is not zero, because if it were zero the co- 
ordinates of.2,, 7, 2; would be proportional to a., 7, 2, and the 
two points MM, M, would coincide and would not determine 
the straight line. The coefficient .A being different from 
zero, the relations above can be solved with respect to a, y, 2, 
and one has for a, y, z expressions of the form 


= Pay Vey Y= BYP VY, 2 = Mey + ve; , 
conversely, whatever v and p be, the point defined by these 
expressions lies on the straight line, since these expressions 


satisfy expression (3). Since a, y, 2 can be divided by the 
same quantity, one can divide them by yp, and on putting 


Y =X one will have 
ph 
= M+ Ay YHYtaAYy, 2=%+ Aq, 


excepting for the point a, y, 2, which corresponds to p = 0. 


INTERPRETATION OF A. — If 2; = z, = 1, one has 


CHAP. XI. HOMOGENEOUS CO-ORDINATES. are 


one recalls the formulas already established (§ 57), and sees, 
on calling M the point (a, y, z), that 


MM, 
MM, 





h=— 


If z, and z, be any quantities different from zero, one could 


divide x, y, 2 by z, and put A 2 = r'; it follows that 


iL 


=D4NS yaDEn 2=1405 
2 


t= p) 
2, a 2) 


one returns, therefore, to the preceding case, and A’ is de- 
fined by 

MM, 

MM, 








moreover, A! differs from A only by the factor * which re- 
; zy 

mains constant when the co-ordinates (a, %, 21), (2 Yo) %)) Pe- 
maining fixed, one imagines that the point m moves along the 
straight line. Whence it follows that the two points 


e= 2, + AD, xe! = %, — AX, 
(M) y=ntrAMy (MH) y=n—- Ay 
.= ral + Zo, e = 21 TER A205 


are harmonic conjugates with respect to the two given points. 
It is easily verified that this result is true if one of the two 
quantities z, or %, for example z,, approaches zero. Then the 
point M', becomes a point at infinity, and the point M is 
the mid-point of the segment determined by the points M and 
M’; one can, therefore, still say that the points M, M' are har- 
monic conjugates with respect to the points M, and MM. If 
the two points M,, M, be at infinity, z, = 2, = 0, it still follows 
that the two points M and M', which are also at infinity, are 
harmonic conjugates with respect to M@, and M. 


ProsieM.— Polar of a point Jf (a, y¥, 2) with respect to a 
conic 


F(X, Y, %) = Av’? + 2 Bey + Cy? +2 Dez +2 Eyz + F?=0. 


378 PLANE GEOMETRY. BOOK III. 


Let M (a, y, z) be a point of the polar: the co-ordinates of any 
point of the straight line M@,M will be 


M+ Av, YW+rY, % + AZ, 


and the values which it is necessary to assign to A in order to 
obtain the co-ordinates of the points where the straight line 
M,M intersects the conic are roots of the equation of the second 
degree : 

SF (a + Aw, + AY, % + AZ) = O, 


S (iy Yu 2%) +A (BP'e, + Ue aes ag Nf (@ Ys %) = 0. 


Let A' and A" be the roots of this equation; the co-ordinates of 
the points M' and M", where the straight line MM, intersects 
the conic, will be 


(M") m+Ala, wW+trAly, 4+A’Z, 
(M") MtA"2, W+Aly, At ANZ; 
since the point M is on the polar, the points M' and M" should 


be harmonic conjugates with respect to M, and M. For this it 
is necessary and sufficient that 


Av =—D!, AT+A"=O; 
that is, 
(5) af. Fly, + Fs, = 9. 
This equation, being satisfied by the co-ordinates of any point 
of the polar, is the equation of the polar. Moreover, one can 
write it 
(5)! ee Me ne nt", ei ZS". = 0. 
If the point M, be at infinity, z, = 0, the polar of this point is 
then the locus of the mid-points of the chords with the angular 


coefficient “; that is, the conjugate diameter of the direction 
vy 
ay, — ya,=0. The equation of this diameter is therefore 


af',+ nt, =9, 


as has been found above. 


CHAP. XI. HOMOGENEOUS CO-ORDINATES. 379 


HomMoGENEOUS CO-ORDINATES OF A STRAIGHT Line. — The 
homogeneous co-ordinates of a straight line whose equation is 


ux + vy + we = Q) 


are the coefficients u, v, w of this equation. Thus, the z-axis 
has the co-ordinates w=0,v20,w=0. The straight line at 
infinity has the co-ordinates u= 0, v=0, w 20. 

Accordingly a linear, homogeneous equation in u, v, w, 


au+bv+cw=0, 


expresses the condition that the straight line (u, v, w) passes 
through the point whose homogeneous co-ordinates are a, b, ¢; 
this equation is called the equation of this point. 

The tangential equation of a curve is the tondition which 
the co-ordinates u, v, w should satisfy in order that the straight 
line with the co-ordinates u, v, w be tangent to the curve; this 
tangential equation will be homogeneous in wu, v, w. Then the 
tangential equation of a circle whose radius is R and whose 
center has the co-ordinates a, b, 1, may be found by expressing 
the condition that the distance from the center to the straight 
line (uw, v, w) is equal to R; this gives the homogeneous equa- 
tion 

(ua + vb + w)? — Rw? + v?) = 0. 
The tangential equation of the conic 

S (a, y, 2) = Ax? + 2 Bry + Cy? + 2 Drz + 2 Eyz + F? =0 

1s au? + 2 buv + cv? + 2 duw + 2 evw 4+ fw? = 0. 


If f = 0, the curve is a parabola, and the tangential equation 
is satisfied by the co-ordinates u=0, v=0, w20 of the 
straight line at infinity: this is what is expressed when one 
says that the parabola is tangent to the straight line at infinity. 

Let (2, 1, Wy), (to, Vg, Wy) be the co-ordinates of two distinct 
straight lines; the equation of their point of intersection will be 


U UV WwW 


380 PLANE GEOMETRY. BOOK III. 


The co-ordinates u, v, w of any straight line D which passes 
through this point can be written 


(D) U=MW+ Aly V=Yy+AVy W= UW, + AW; 


these formulas may be verified in the same manner as for- 
mulas (4). If the sign of A be changed, the co-ordinates of 
a second straight line are found to be 


Dd.) u! ~-s Uy raz! AUd, y! act UV) or AVo, w! = Wy, = AWs, 
which is the harmonic conjugate of the first with respect to 
the two given straight lines; for, the equation of the straight 
line D is 

UL VY + WZ = WX + VY + We +A (Ue + VY + WZ) =), 
and that of D' is, similarly, 

Ue + VY + WE — A (Uge + VY + WR) = 0, 

which proves the theorem (§ 69). 


Exercisr. — Prove that the pole of the straight line (u,, v1, Wy) 
with respect to a curve of the second class whose tangential equa- 
tion is p(u, v, w) = 9, ts given by the equation 


ug'y, + Voy, + Wh'w, = 9, 
ud'y + vidb'y + wid'w = 0. 


or 


TRILINEAR CO-ORDINATES. 
331. Derinition. — Consider three linear equations 
a= ax + by + c, 
(6) B=a'e+ bly+c'z, 


y a al'e + b'y + C2, 
where the determinant 


CHAP. XI. TRILINEAR CO-ORDINATES. 381 


is different from zero. To every system of values of 2, y, z in 
these equations there corresponds a single system of values of 
a, B, y, and, conversely, to every system of values of a, B, y a 
single system of values of a, y, z. 

If x, y, 2 be the homogeneous co-ordinates of a point of a 
plane, it follows also that to every system of values of «, B, y, 
all of which are not zero at the same time, there corresponds 
a definite point M, and to every point M of the plane there 
corresponds a unique system of values of «, B, y, with the con- 
dition that systems such as «, B, y, and pa, pB, py, are not to 
be regarded as different systems. 

The quantities «, 8, y, are called the trilinear co-ordinates of 
the point M with respect to the triangle of reference whose sides 
have the equations 


ax + by+cz=0, ae+bly+tez=0, alea+o"y+cl'z=0. 


GEOMETRIC INTERPRETATION. —If one take z=1, the tri- 
linear co-ordinates « are equal to the distances of the 
» Ps ¥ q 
point M from the three sides of the triangle of reference 
multiplied by factors which have the same sign when the point 
M varies. In particular, if one consider the equations 
? 


a =—(xcosa+ ysina — pz), 
(7) B=—(«xcosb+ysinb —qz), 
y=—(xcose +ysine — 72); 


and if one suppose the origin of Cartesian co-ordinates to be 
within the triangle, one sees that for z= 1, a, B, y are equal 
to the distances of the point M from the sides affected with 
proper signs. This sign is + for a side AB of the triangle 
of reference when the point M under consideration and the 
vertex C opposite to AB are situated on the same side of AB; 
it is — in the contrary case. 

In order to find in trilinear co-ordinates the equation of a 
given curve in homogeneous co-ordinates, f(a, y, z)= 0, we 
replace a, y, z by the values found by solving equations (6) with 
respect to a, y, z The values obtained for 2, y, z being homo- 


382 PLANE GEOMETRY. BOOK III. 


geneous and linear in «, f, y, the new equation F(a, 8, y)= 0 
will be homogeneous in «@, B, y, and of the same degree as f. 
Conversely, if one be given an equation F(a, B, y)= 9, in tri- 
linear co-ordinates, it will be sufficient to replace «, B, y by 
expressions (6) in order to have the equation of the curve in 
homogeneous co-ordinates. 

Let, for example, 


S(&, Y, 2) = ux + vy + wz = 0 


be the equation of a straight line in homogeneous co-ordinates ; 
if the preceding substitution be made, one will get for the 
equation of this same straight line 


F=Ua+ VB+Wy=0. 


One can return to equation (7) by replacing a@, B, y by their 
values (6). It is evident also that 


u=aU+a'V+a'W, 
(8) v=bU+0'V+b'W, 
w=cU+ec'V+c"W. 


The coefficients U, V, W of equation F are called the tangential 
co-ordinates of the straight line in the new system; equations 
(8) express the homogeneous tangential co-ordinates (wu, v, w) 
as functions of the new (U, V, W), and, conversely, they make 
it possible to transform every homogeneous tangential,equation 
in u, v, w, o(u, v, w)=O0, into another of the same degree 
®(U, V, W)=0, and conversely. 


Toe EqQuaATION OF THE STRAIGHT LINE AT INFINITY IN 
TRILINEAR Co-oRDINATES. —The co-ordinates of a point at 
infinity have been defined as a system of values 2, y, z, in 
which z is zero. If formulas (6) be solved with respect to 
z, and if the homogeneous linear expression in @, 8, y found 
for z be equated to zero, one obtains a condition which is called 
the equation of the straight line at infinity. For example, one 
deduces from equation (7), 


zD=«asin(b —c)+Bsin(e — a)+ ysin(a — 8), 


CHAP. XI. TRILINEAR CO-ORDINATES. 383 


where D is a constant factor; whence one obtains for the 
equation of the straight line at infinity, 


asin d+ fsinB+ ysinC=0, 
where A, B, C are the angles of the triangle of reference. 


Ture EquaATION oF A STRAIGHT LINE PASSING THROUGH 
Two Pornts.— Let M, and M, be two points whose trilinear 
co-ordinates are (4, Bi, y1)> (G2, Bo y2), then will the equation 
of the straight line which passes through two points be 


a B oy 
(L) By yy — 0. 
lo By Yo 


The co-ordinates of a point M of the straight line may be 
expressed by the formulas 


(M) = +A, B=Bit+ABy Vay 1 Aye 
where A has the same meaning as in formulas (4). In fact, 
call (21, Yiy 21) (Xap Yo) 22), (@, y, 2) the homogeneous co-ordi- 
nates of the points My, @,, M. One has, according to (6), 

Oy = A, + DY, + Cy Ly = AX, + DYy + Cy H= aH + bY + C2; 


since Xx — xy + 2, ois WO 


one has also « = a, + Ad, and similar expressions for f and y. 
Whence it follows that the point M’ with the co-ordinates 


(M") , — Alley Bi — APs, 4 eae Ay2 
is the harmonic conjugate of M with respect to the segment 
MM. 

By a calculation similar to that which precedes equation (5), 
one can show that the polar of the point (@, #1, yi), with 
respect to the conic 

F(a, B, y)=AC+A'B+A"y4+2 BBy+2 Blya+2 B"aB=0, 
has the equation 


oF", + BiF's + Fy =0, or ak", + BF's, + Fy, = 0. 


384 PLANE GEOMETRY. BOOK III. 


TANGENTS IN TRILINEAR CO-ORDINATES. — Let F(a, B, y)=0 
be the equation of a curve, and @, B,, y, be a point situated 
on the curve having the homogeneous co-ordinates 2, %, 2). 
On replacing a, B, y by their values (6), the equation of the 
curve in homogeneous co-ordinates will be obtained: by this 
substitution F(a, B, y) is transformed identically into f(a, y, 2), 
the first member of the equation of the curve in homogeneous 
co-ordinates. The equation of the tangent at the point a, y,, 2 1s 


het gece 2) ap 
J(&, Y; Z) = F(a, B, Y) 


gives by reason of equations (6), 


or the identity 


Je Oia ot par, 
=—0F +r +0 2". 
yy Ga By v1 
Ue COR are re al ee 
whence af, + Uf'y, + 2's, = Cl", + BP's, + yi. 


The equation of the tangent at the point @, £1, y, 1s there- 

fore 
al”, + BF’, + yF,, = 9. 

The tangential co-ordinates of this tangent are given by the 

equations 
pC Es re F's» fo fea Fy 

p being a constant different from zero. 

One can demonstrate in a similar manner that: 

1° The equation of the point of intersection of the two 
straight lines D, and D,, whose co-ordinates are (U;, Vi, Wi) 
and (Ux Vy W,), is | 


9 recs aes | 6 
U, Vi WwW, |=9; 
Ue a Ve 


2° The co-ordinates of any straight line D passing through 
this point are 


(D) U=U,+A(Uy V=VitrVn WH Wi 40M; 


CHAP. XI. TRILINEAR CO-ORDINATES. 385 


3° The straight line D', whose co-ordinates are 
(D') U, — AU), Vi —ADV;, W, —AW,, 
is the harmonic conjugate of D with respect to the two straight 
lines D, and D,; | 

4° If U,, Vi, W, be the co-ordinates of a tangent to the 
curve whose tangential equation is ®(U, V, W) =0, the equa- 
tion of the point of contact is 

Ue!, + VO'l, + Waly =, 
and the co-ordinates of this point are given by the formulas 
pa = $',,, pp = ®',, py = P'y,. 

Below are given some very simple applications of the pre- 

ceding considerations. 


Exampue I.— Form the general equations of conics conjugate 
with respect to the triangle of reference. 

Let 
F(a, B, y)= Ae + A'? + A"y +2 BBy +2 B'ya+2 BaB = 0 
be the equation of such aconic. The polar of each vertex of 
the triangle of reference with respect to this conic ought to be 
the opposite side. The polar of the point («, 8), y:) being 


oF" + B\F's ae vil’, = 0, 
that of the vertex of the triangle whose co-ordinates are @, = 0, 
i, = 0, yi 29, is 

4 F', is Bla+ BB+ A"y=0; 

this polar should coincide with the opposite side y= 0, and 
hence B= B'=0. Similarly it may be shown that B"=0, 
and the required equation will be 

Ad’ + A'? + Al"y = 0. 


The tangential equation of these same conics is 


It should be noticed in these formulas that one can suppose 
that one of the vertices or one of the sides of the triangle of 
reference is at infinity. 

2B 


386 PLANE GEOMETRY. BOOK IIt. 


General equation of conics inscribed in the triangle of refer- 
ence. The most general curve of the second class has in 
tangential co-ordinates the equation 


AU?+ A'V?+ A"W?4+2BVW+2 B'WU+2 B"UV=0; 


on expressing the condition that this curve is tangent to the 
three sides of the triangle of reference whose co-ordinates are 
respectively 


V=—O0 and W=—0,,W=—0 and 1) —0, U-—0 and-7 0; 


it follows that A= A'=A"=0. The tangential equation of 
the conics in question is therefore 


BVW+ B'WU + B"UV = 9 


Moreover, the equation in trilinear co-ordinates is 








Ob ee 
BO he, re 
BOB 0 ay 
on ye 


or B’a?+ B"Q?+ B'"y?—2 B'B"By—2 BB"ay—2 BB'aB=0. 


The general equation of conics inscribed in a quadrilateral. 
Let, in homogeneous co-ordinates, 


—lat+ly+l'z=0, Q=mr4m'ytm'e=0, 
R=netn'yt+n'2=0, S=ka+k'y+kh'z=0, 


be the equations of the four sides of the quadrilateral. Three 
homogeneous linear functions, 


a=ar+a'yta'z, B= ba + bly + bz, y=cutcly+ cle, 


and four constants p, q, 7, 8; can always be found such that one 
has identically 


(1) pP=a+Bt+y, IQ=«—-B-Yy, 
rR=a—Bty, sSS=a+B—y; 


CHAP. XI. TRILINEAR- CO-ORDINATES. 387 


the identification of the two members of these identities will 
give twelve homogeneous equations of the first degree con- 
necting the thirteen unknown constants 


i att. ae Ue i all. . 
a,a', a"; 6, bY bY"; ec e's pig, 7, s. 


The calculation can be simplified as follows. The following 
equation may be deduced immediately from identities (1): 


(2) pP+qQ-—rR—sS=0, 


which will determine p, q, 7, s, or, better, the ratios of any 
three to the fourth. The constants p, g, r, s being thus de- 
- termined, it follows that one of the identities (1) is deducible 
from the other three; one can write 


Z2a=pP+qQ=rR+s8, 


2B=pP—rR=sS — qQ, 
2y=pP—sS =rR—qQ; 


the functions @, B, y are therefore known; the two expressions 
found for each of these linear functions are identical by reason 
of the identity (2). The three straight lines e=0, B=0, y=0 
are the diagonals of the given complete quadrilateral. In 
fact, the equation of the straight line « =0 can be written in 
either of the forms 


pP+qQ=0, rR+sS=0; 


the first shows that this straight line passes through the vertex 
of the quadrilateral which is the intersection of the sides 
P= 0, Q = 0, and the second that it passes through the oppo- 
site vertex R= 0, S = 0. 

Thus, on choosing as the ae of reference the triangle 
formed by the three diagonals of a complete quadrilateral, the 
equations of the sides of the quadrilateral can be written in 
the form | 


at+Bt+ty=0, «+B—y=0, «e—B+y=0, e—B—y=0. 


The vertex P=0, Q=0 has the trilinear co-ordinates « = 0, 
B++y=0, and the opposite vertex R=0, S=0 has the tri- 


388 PLANE GEOMETRY. BOOK III. 


linear co-ordinates «= 0, 8B=y. The first of these points has 
the tangential equation V+ W=0, and the second V— W=0: | 
the ensemble of these two opposite vertices is therefore repre- 
sented by the tangential equation V?—W?’=0. Similarly, 
the ensemble of the two opposite vertices P= 0, S=0, and 
Q=0, R=0 is represented by the tangential equation 
U2—V?=0. The general tangential equation of conics in- 
scribed in a quadrilateral is, therefore (§ 307, Ex. IL.), 


V?—W?242r(U?— V2)=0, 
or AU? ++(1 —A)V?— W? = 0. 


The equation of the same system of conics in trilinear co-ordi- 
nates 1s | 


Vie B° 
dete eit oda eo 
seas ene 


Remark. — On putting 


-1 
a=, p= ’ y=2= 1, eS, 





one obtains the equation 


a? 


ay? 
eee 
pp pe 





of confocal conics. 

Examp.Le II.—Let us consider two reciprocal polar tri- 
angles with respect to a given conic; for simplicity take the 
sides of one of the triangles as lines of reference fulfilling 
the definition of new co-ordinates, and let 


f(a, B, y) =4 (Ae? + AR? + A" +2 BBy + 2 B'ya 
+2 B"aB) =0 

be the equation of the conic. The polar of any point (a', B', y') 
has the equation «f+ B'f'p+yf'y=90. In particular, the 
polars of the three vertices (B'=0, y'=0), (7'=0, a’ = 9), 
(a' = 0, B'=0) of the triangle have the equations i 

f'g = Aa + BB + By, f'g = Bla + A'B+ By =9, 

fly = Blu+ BB+ A'y =0. 


CHAP. XI. TRILINEAR CO-ORDINATES. 389 


These polars are the sides of the second triangle. The co- 
ordinates of the point of intersection of two corresponding 
sides a=0, f',=0 satisfy the equations «=0, B'B + B'y=0, 


or a= 0, te 10; this point is situated on the straight 
line at P+ 0, and similarly for the other two sides. 


Thus the three points of intersection of the corresponding sides 
of two reciprocal polar triangles lie on a straight line. 

A vertex of the second triangle being given by the two equa- 
tions f',=0, f'g=0, the straight line Bf. = Bip passes 
through this point; since the equation does not contain the 
letter y, this straight line passes through the vertex (« = 0, 
B = 0) of the first triangle. The straight lines which join the 
corresponding vertices being represented by the equations 
Bf', = B'f's = B"f',, it follows that these three straight lines 
pass through a common point. 


Exampie III. — A triangle abe is inscribed in a conic; two 
of its sides ab and ac revolve about two fixed points p and g 
(Fig. 205); find the envelope of the third side bc. Let y=0 
be the equation of the straight 
line pg, «= 0 and B=0 those 
of the tangents at the points 
d and e in which this straight 
line intersects the curve; the 
equation of the conic will have 
the form aB—y’=0. The 
points p and q may be regarded 
as the points where the straight 
line y = 0 is intersected by the 
two straight lines «+ pB = 0, 
a+gB8=0, which pass through 
the point of intersection o of 
the tangents at d and e. Any point a of the curve can be 
determined by the intersection of two straight lines «—ay=0, 






Fig. 205. 


B- a = 0, which pass through the points d and e, a being an 


arbitrary parameter which defines the position of the point 


390 PLANE GEOMETRY. BOOK IIl. 


a on the curve. On assigning to this parameter another 
value 6, one obtains another point b. Any straight line 
passing through the point a has an equation of the form 


a—ay+k (6 = X) = = 0; in order that this straight line pass 
through the oe 6 which is represented by the two equations 
a—by=0, B— Z = 0, it is necessary to make k=ab; thus 


the straight line which connects any two points a and b of the 
curve has the equation « + abB — (a +b) y = 0. 

Let now a, b, ¢ be the values of the parameter for the three 
vertices a, b, c of the triangle; since the side ab passes through 
the point p, one has ab =p; since the side ac passes through 
the point g, one has similarly ac=q; the side be has the 
equation a +beB —(b+c)y=0; if b and c be replaced by 


their values E and , the equation becomes 


wa + pgB —(p+q)ay=0. 


If the variable parameter a be eliminated between this equa- 
tion and the following equation, 


2au—(pt+q)y=9, 


which is obtained by equating to zero the derivative of the 
preceding equation with respect to a, one obtains the equation 
of the envelope of the straight line bc, ¢ 


c 


This envelope is a conic which touches the first at the points d 
and e. 

The tangential co-ordinates U, V, W of the variable straight 
line are given by the equations, 


pU=a’, pV=pq, pW=—(p+ qa; 


the elimination of a and p gives the tangential equation of the 


envelope, 
UV (p+ 4)*— W*pq =0. 


CHAP. XI. TRILINEAR CO-ORDINATES. 391 


If tangents be drawn to the proposed conic at the points 
a, b, ¢, a circumscribed triangle a'b'c! is formed, of which the 
two vertices b' and c! slide on the two fixed straight lines P 
and Q, polars of the point p and q; the curve described by the 
vertex a', pole of the straight line be, is the reciprocal polar of 
the envelope; therefore, it is also a conic with a double contact 
with the first along the line de. 


EXERCISES. 


1. The eight points of contact of the tangents common to 
two given conics are situated on a conte. 

2. A triangle is inscribed in a conic; two of its sides pass 
through two fixed points or revolve on two conics doubly tan- 
gent to the first; the envelope of the third side is a conic. — 
The converse theorem. : 

3. A polygon with n sides is inscribed in a conic; n—1 
sides revolve about conics doubly tangent to the first; the 
envelope of the nth side is a conic. — The converse theorem. 

4. Two conics S and S' are given, and also two tangents to 
the conic S'; the six straight lines which jom two by two the 
four points in which these tangents intersect the conic S are 
two by two tangent to the same conic which passes through 
the point of intersection of the conics S and S'.— The converse 
theorem. 

5. Being given three conics which have four given points 
in common, a triangle inscribed in one of them has two of its 
sides tangent respectively to the other two conics; the third 
side envelops a conic. — The converse theorem. 

6. Being given n conics which have four given points in 
common, a polygon of n sides inscribed in one of them has 
n —1 of its sides tangent respectively to the other conics; the 
nth side envelops a conic. — The converse theorem. 

7. A polygon in one of its positions is inscribed in a conic 
and circumscribed about another conic; if a vertex be made to 
move on the first conic, in such a way that n —1 sides are tan- 
gent to the second conic, the nth side will always be tangent to 
this second conic. eee 


392 PLANE GEOMETRY. BOOK TIt. 


The conics which have four common points in Theorems IV., 
V., VI., and VIL., can be replaced by homothetie conics having 
two common points, and in particular by circles having in 
pairs the same radical axis. 

8. The envelope of the straight lines which intersect two 
given conics in four points, which are harmonically arranged, 
is a conic. — The converse theorem. 

If the two conics have in trilinear co-ordinates the equations 
w+ B+ y=0, Aw’ + BB’? + Cy? =0, the necessary and suffi- 
cient condition in order that the straight line wa + vB + wy =0 
intersects the two conics in four points, which are harmonically 
arranged, is 


(B+ C)w + (C+ A)v?+(A+ Bw? =0. 


What will happen when 4+ B=0? Apply the result to 
_ the particular case where one of the conics is a circle, and the 
other an equilateral concentric hyperbola. 

9. We know that the polars of a point p, with respect to 
all of the conics which have four points in common, pass 
through a fixed point g; if the point p describe a straight line, 
the point q describes a conic. — The converse theorem. 

10. If two sides of a triangle inscribed in a conic revolve 
about any two given curves, the third side envelops a third 
curve; show that the straight lines which join the vertices of 
the triangle to the points of contact of the opposite sides pass 
through a fixed point. — The converse theorem. i 

11. Being given a hexagon inscribed in a conic, the points 
of intersection of the opposite sides are located, and also the 
points of intersection of each of three diagonals with the two 
opposite sides; the nine points thus determined are situated on 
three straight lines which pass through a fixed point. 

12. A conic S is given, and a variable conic S’ is con- 
structed, which intersects the first in two fixed points and 
which touches two fixed straight lines whose point of inter- 
section is situated on the conic S; the envelope of the 
straight line which passes through the other two points of 
intersection of the conics S and S' is a conic. — The converse 
theorem. 


CHAP. XI. TRILINEAR CO-ORDINATES. 393 


13. A quadrilateral is circumscribed about a conic; if any 
tangent be drawn to the conic, one knows that the ratio of the 
product of the distances of this tangent from the two opposite 
vertices of a quadrilateral to the product of the distances of 
this same tangent from the other two opposite vertices is con- 
stant. Show that this ratio is equal to the product of the 
distances of the first two vertices from one of the foci divided 
by the product of the distances of the other two vertices from 
the same focus. 

14. The six sides of any two triangles inscribed in the same 
conic are tangent to another conic. — The converse theorem. 

15. Three points are said to be conjugate with respect to 
a conic, if the polar of one of them is the straight line which 
joins the other two; show that the two systems of three con- 
jugate points with respect to a conte are situated on another 
conic. — The converse theorem. 

16.. Find the necessary and sufficient conditions in order 
that a conic coincide with its reciprocal polar with respect to 
the circle v7 +7? -1=0. 

17. The tangential equation of a conic in rectangular co- 
ordinates being 


aw? + 2buv + cv? +2du+2e4+f=0, 


show that the circle which is the locus of the vertices of the 
right angles circumscribed about the conic, has the equation 


f(e’?+ yy’) —2d~a—2Zeyt+a+c=0. 


18. Consider a variable conic tangent to four fixed straight 
lines.. Show that the circle which is the locus of the vertices 
of the right angles circumscribed about this conic passes through 
two fixed points, real or imaginary. 

19. Consider a variable conic tangent to three fixed straight 
lines. Show that the circle which is the locus of the vertices of 
the right angles circumscribed about this conic is the orthogonal 
conjugate to the triangle formed by the three straight lines. 

20. Consider a conic whose equation is 


S(&, y) — (la + my +n)’ = 0, 


394 PLANE GEOMETRY. BOOK TI. 


S(@, y) being a polynomial of the second degree in @ and y. 
Show that the tangential equation of this conic is 


(Wu, 0) — Qu +p $y) =0, 


¢@ being of the second degree in w and v. Apply the result 
to the case where 


SF (@, y) = (@ — a)’ + (y — bY’. 


21. Form the tangential equation of the curve generated 
by a point of a circumference rolling within a circumference 
of triple radius. (Hypocycloid with three cusps.) On writing 
the equation of the tangent to the curve in the form 2 sin « — 
ycosa@=p, show that p has the form p=acos(8a+ @), a 
and m being constants. 

22. Find the envelope of the axes and the tangents at the 
vertex of parabolas inscribed in a triangle. 

23. Find the envelope of the axes of equilateral hyperbolas 
circumscribed about a triangle. 

24. Show that the necessary and sufficient condition in order 
that two straight lines whose co-ordinates are (u, v), (w', v’) 
intersect on a conic 


Aa’? + 2 Bey + Cy? +2 De+2 Ey+F=0, 


is 
ve aed 6 ery B ae ges 
Ds eae Souhe usa ees oy 
SO ee ee nT : 
1 iO A) 
Coe aoe 








CHAP. XI SECANTS COMMON TO TWO CONICS. 395 


CHAPTER XII* 
SECANTS COMMON TO TWO CONICS. 


332. We have found above (§ 286) an equation of the third 
degree 


(1) A+ @d + O24 A = 0, 
giving the value of A, for which the equation 
S+AS'=0 


represents two straight lines. It is proposed to investigate the 
nature of the roots of equation (1), called equation in A, and 
to study the nature of the points common to the two conics 
S=0, S'=0. We adopt for this purpose a method due to 
M. Darboux, which is reproduced in the excellent Treatise 
on Analytic Geometry by M. Pruvost. This method is based 
on the following lemmas. 
(1) If the equation 


S = Aa? +2 Bry + Cy? +2 Dez"+ 2 Eyz + F2 =0 


represent two parallel or concurrent straight lines, the homo- 
geneous co-ordinates a, b, c of the point of intersection, situated 
at an infinite or finite distance, satisfy the conditions 


00. 2b ae bee 
(2) fb Pa bee 
a 
In fact, let 
P=vuze+rvyt+uz=0, Q=u'e+v'y+wz=—0 
be the equations of two straight lines; one will have identically 


S= PQ; 


396 PLANE GEOMETRY. BOOK Itt. 


whence, taking successively the partial derivatives of the two 
members with respect to a, y, z: 


Ax + By + Dz=3(Pu' + Qu), 
Bu + Cy + Ez =4(Pv' + Qv), 
Dx + Ey + Fz =} (Pw'+ Qu). 


The homogeneous co-ordinates a, b, c of the point of inter- 
section of the two straight lines reduce to zero the first mem- 
bers P and Q of the equation of the two straight lines; they 
satisfy, therefore, the three equations 


Aa+ Bb+ Dce=0, 
Ba+ Cb + Ec =0, 
Da+ £b+ Fe =0; 


whence may easily be deduced 


0c 
Pe Ve 
a_b_¢ 
boc 
a ee Ge 
d 6 f J 


on multiplying the first group of these relations by a, the 
second by b, the third by ¢, one obtains relations (2), which 
was to be proven. 


Depuctions. — If the equation S = 0 represent two straight 
lines, discriminant A is zero, and equation (1) has one of its 
roots equal to zero. If one have further ® = 0, this root zero 
is double; and then the point of intersection of the straight 
lines P=0, Q=0, which constitute the conic S, lies on S’. 
In fact, one has 


®@= A'at+2Bb4+C'c+2 D442 He+ Ft. 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 397 


Owing to relations (2), the condition © = 0, in which a, b, ¢, 
ete., are replaced by the proportional quantities a’, 2 ab, b*, --, 
becomes 

Al? +2 Blab +C'? +2 D'ac+ 2 E'be + Fic’ = 0, 


the condition required that the point (a, }, c) lies on the conic 
S' = G. 

(2) If, besides the conditions A = 0, ® = 0, one have further 
@'—0, one of the straight lines into which the conic S= 0 is 
decomposed is tangent to S'. In this case equation (1) has a 
triple root zero. 

The identity 

S = PQ =(ux + vy + wz) (ula + v'y + w'2) 
gives 
A=uu', 2B=w'+vu', C=v0', 


2D=wuw'+wu', 2E=vw'l+w'!, F=ww'; 
and, since 

@'= Aa'+ 2 Bb'+Cc'+ 2 Dd'+ 2 He'+ Ff, 
the condition ©’ = 0 becomes 

aluu' +b! (uv! + u'v)+ eve! + d! (uw! + wu') 
+e'(vw' + v'w)+ fiw! = 0; 

this condition shows that the pole of one of the straight lines 
P=0, Q=0 with respect to S' lies on the other (§ 296), that 
is, that these two straight lines are conjugates with respect to 
the conic S'=0. Since their point of intersection is situated 


on this conic, one of these straight lines is necessarily tan- 
gent to it. 


333. We have to examine what happens in case the conic 
S =0 consists of two straight lines; we occupy ourselves first 
with the general case. For this purpose, suppose that S and 
S' are any conics whatever, and call A, a root of equation (1). 
The conic 

S+S'= 0 


will be decomposed into two straight lines P, = 0 and Q, = 0. 


398 - PLANE GEOMETRY. BOOK III. 


Put . * 
(3) S+r,S'=S, oS + BS'=S', 


«and B being any two constants subject to the condition that 
they will not reduce the quantity (8 —«d,) to zero, which 
determines the coefficients of S and S' in relations (3). Then 
S,=0 is the equation of the pair of secants common to the 
conics S = 0, S'=0, which correspond to the root A= )A,, and 
S',=0 is the equation of any conic, distinct from S,=0, 
which passes through the points of intersection of the given 
conics S=0, S'=0. Since identities (3), solved with respect 
to S and 8%, give 


(B oe (X,) S = BS, oa AG a 
(B — ard,)S'=—aS,4+ 84, 
the general equation S + AS' = 0 can be written 


Si (8 —— ar) + S'(A — Aj) —- 0, 


or : S,+ pS) = 0, 
on putting 

A—A 
4 oe oh 
(4) ae ery? 


Thus, » being connected with A by relation (4), the two equa- 
tions 

S+AS'=0, S,+pS',=0 
represent the same conic. If one seek the value’ of p for 
which this conic reduces to two straight lines, one obtains an 
equation of the third degree in p: 
(5) A, + Oy + Oly? + Alp? = 0, 
in which A, = 0, since S, is decomposed into two straight lines. 
It follows from relation (4) that to the root A = A, of equation 
(1) there corresponds the root »=0 of equation (5) with the 
same degree of multiplicity. 

If the root A, be simple, the root »=0 is also simple; ©, is 
therefore different from zero, and the point of intersection of 
the straight lines represented by equation S, = 0 is not on the 
conic S'. 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 399 


If the root A, be double, the root » = 0 of equation (5) will 
be double ; @, will be zero, and the point of intersection of the 
two straight lines S,;=0 will be on the conic 8. It can 
happen, as a particular case, that the minors of A, are zero; 
then @, is zero whatever the conic S’; may be; the conic S, = 0 
reduces to a double straight line. 

Finally, if the root A, be triple, the root »=0 of equation 
(5) will likewise be triple; ©, and @', will be zero; the point of 
intersection of the straight lines S, = 0 will be on S', and one 
of these straight lines will be tangent to S'. It can happen, 
as a particular case, that all the minors of A, are zero; then 
the conic S, = 0 reduces to a double straight line tangent to S',. 

Call Ay, As, As the roots of equation (1) in X. 

1° If these roots be unequal, each pair of secants consists of 
two distinct straight lines, whose point of intersection does not 
lie on any of the given conics. These conics intersect there- 
fore in four distinct points. 

In order to learn how many of these points are ‘real, one has 
recourse to the following considerations : 

(a) The three roots Aj, A», As are real, and the pairs of secants 
corresponding to two of them are real. These two pairs of 
real straight lines intersect in four real and distinct points. 
These four points of intersection of the conics are, therefore, 
real and distinct. It is plain that the third pair of secants is 
also real, since it passes through the points of intersection of 
two conics which are real. 

(b) The roots Ay, As Ag are real, but one pair only of the 
secants is real. Then the four points of intersection are imagi- 
nary. In fact, a pair of imaginary secants with real coefficients 
have in common but one real point, the point of intersection of 
the secants; since this point does not belong to any of the 
conics S or S', the points of intersection, all four of which are 
situated in this imaginary pair, are necessarily imaginary. 

(c) One root A, is real, the other two A, and A; are imaginary. 
Two of the points of intersection are real and two imaginary. 
If A, be imaginary, A, = p + iq, the corresponding pair 


S +(p + ig)S'=0 


400 PLANE GEOMETRY. BOOK III. 


is composed of two imaginary straight lines, but not imaginary 
conjugates. In the first place, this pair is composed of two 
imaginary straight lines; because if one of these straight lines 
were real, the co-ordinates of a point of this straight line would 
reduce S + pS'to zero on the one hand, and the coefficient S' of 
i on the other; that is, S and S'; the two conics would have, 
therefore, a common straight line, which contradicts the 
hypothesis. Moreover, the pair considered does not consist 
of conjugate imaginary straight lines; because, if that were 
so, the point of intersection of these straight lines would be 
real, and its co-ordinates would reduce to zero, simultaneously, 
the partial derivatives of S+(p + tg)S' with respect to a, y, 2, 
that is, the three partial derivatives of each of the polynomials 
S and S'; the two conics S=0, S'= 0 would consist of two 
pairs of concurrent straight lines, which contradicts the 
hypothesis. The pair corresponding to the root A, will be 
composed of two straight lines whose equations are 


P+iQ=0, R+iS=0; 
and that corresponding to the root A, of the two straight lines, 
P—iQ=0, R-iS=0. 


These two pairs and, consequently, the two conics intersect 
therefore in two real points, the one situated at the intersection 
of the straight lines P= 0, Q = 0, the other at the intersection 
of the straight lines R=0,S=0. The other two points of 
intersection are imaginary, because on one imaginary straight 
line there can be but one real point. 

. 2° If the equation in A have a double root A, and a simple 
root A, the point of intersection O, of the two assumed distinct 
straight lines of the pair 


NW iss 0 es 


corresponding to the double root, lies on the conics S and S', 
and, more generally, on all of the conics #S + BS'=0; the 
point of intersection O, of the straight lines of the couple, 


S,= 5 +A,S8' = 90, 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 401 


does not, on the contrary, lie on any of these conics other than 
S, itself. In particular, the point O, lies on the pair S;, whilst 
O; does not lie on the pair S;. The points of intersection of 
the two pairs of straight lines S,; = 0, S;= 0, and, consequently, 
those of the two conics are therefore disposed in the following 
manner: two of these points are coincident with O,, situated on 
that straight hne D of the straight lines of the couple S;, 
which passes through this point; the other two points A and 
B are at the intersection of the couple S, with the second 
straight line D' of the couple S,. The two conics S and S’ are 
tangent in O, to the straight line D, and intersect in the two 
points A and B situated on D’. 

Since the couple S,; = 0 is composed of two distinct straight 
lines, their point of intersection O,, that is, the point of con- 
tact of two conics, is real; the pair S, containing the tangent 
to these conics at the point O, is real; the points A and B 
will be real or imaginary according as the couple S, is real 
or imaginary. — 

If the pair S, be a double straight line, the points of inter- 
section of the conics are coincident, two and two, with the 
points where the double straight line intersects the two straight 
lines D and D' of the pair S; corresponding to the simple root. 
The two conics have a double contact: the double straight line 
is the chord of contact, and the two straight lines D and D' 
are the tangents at the points of contact; these points of con- 
tact are real or imaginary according as the pair S, is real or 
imaginary. 

3° The equation in X has atriple root A,. If the pair which 
corresponds to this root consist of two distinct straight lines, 
their point of intersection O les on the two conics, and one of 
the straight lines of this pair is tangent to the two conics at O. 
The two conics intersect therefore in three points coincident 
with O and in one other point A; these points O and A are 
necessarily real: In this case, it is said that these two conics 
osculate each other in O. 

If the couple which corresponds to the triple root A be a 
double straight line, this straight line ought to pass through 
the points common to the conics and be tangent to the two 

2c 


402 PLANE GEOMETRY. BOOK III. 


conics; it will be tangent to the two conics at the same point O. 
These two curves intersect therefore in four points coincident 
with O; they are said to have a contact of the third order 
or to be sub-osculatory. 

If the pair corresponding to the triple root A, be indeter- 
minate, the two conics are coincident. 


Remark. — Determine the conditions for which the equa- 
tion in d is indeterminate; it would be necessary for this that 
all conics represented by the equation 


S+ AS'=0 


decompose into systems of straight lines. One will have, 
therefore. 
A= 6 = 6 — Al 0 

The conditions A=0, A'=0 show that the two conics be- 
come systems of straight lines. The condition © =0 shows 
that the point of intersection O of the straight lines repre- 
sented by S=0 is on S'=0, and @'=0 shows similarly that 
the point of intersection O' of the straight lines S'= 0 is on 
S=0. If the points O and O' be distinct, it follows from 
the preceding that the straight line OO' ought to belong to 
the two conics; if O coincide with O', the two conics consist 
of pairs of straight lines intersecting in the same point. Con- 
versely, if one of these conditions be fulfilled, the equation 
in A is indeterminate. One has, in fact, A=@=@0'=A'=0. 
It can be verified that, in these two cases, the equation 


S + ,AS'=0 


represents straight lines whatever A may be. In fact, if the 
conics S and S' be systems of straight lines having a common 
straight line, one has identically 


S = PQ, S'= PR, 


P, Q, R being linear functions in 2, y, z, and the equation 


S+AS’=0 becomes 
P(Q+AR) =9, 


which represents two straight lines. 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 403 


If S and S' be systems of straight lines intersecting in the 
- same point, one has 


S= PQ, S'=(aP+bQ)(a'P +0'Q); 


S+A8S' will then be a homogeneous polynomial of the second 
degree in P and @ and is consequently, whatever may be, 
resolvable into two factors of the first degree : 


(@P + BQ) (a'P + B'Q). 


EXAMPLE. — Consider a conic which, referred to rectangular 
axes, has the equation 


S = Aa’ + 2 Buy + Cy? +2 Hy=0; 


that is, a conic tangent to the a-axis at the origin; find the 
radius of the circle osculating this conic at the origin. 
The osculating circle, being tangent to the x-axis, will have 


the equation 
Dar yp 2 hy = 0. 


The equation S +A S'= 0 will therefore be 
(A+A)a? +2 Boy + (C+) y+ 2(E+AR)y =0. 


On equating the discriminant of the polynomial to zero, the 
following equation in J is obtained : 


(A+A)(Z+ dR)? =0, 
which has, whatever R may be, the double root A, = — , and 


the simple root A;3=— A. In order that the two conics oscu- 
late each other, it is necessary and sufficient that the equation 
r has a triple root; that is, that A; = A, or 


E E 
—=A, R=—. 
yi : al 


The ordinate of the center of the osculating circle is there- 


fore =, and the radius of the circle is the absolute value 


of E This radius is called the radius of curvature of the 


conic at the origin. 


404 PLANE GEOMETRY. BOOK III. 


If we suppose 


the circle is the osculating circle; in order that it be sub- 
osculatory, or have a contact of the third order, it is necessary 
and sufficient that the pair of secants corresponding to the 
triple root 


be a double straight line. This couple is 
y’(C — A) + 2 Bry =0; 


in order that it be a double straight line, it is necessary and 
sufficient that B=0. Whence the equation of the conic 


becomes 
Av’? + Cy? + 2 Ey = 0, 


and the origin is at a vertex. It is therefore only at the 
vertices of a conic that the osculating circle becomes sub- 
osculatory. 


THEOREM. — The roots of the equation in X remain the same 
when the two conics S and S' are referred to other co-ordinate 
axes. 


In fact, suppose that the two conics S=0, S'=0 be referred 
to new co-ordinate axes O,2,, Oy. The formulas for making 
this change of co-ordinates are 


% = AX, + bY, + CA, 
(6) y= a'e, + bly, + eZ, 
a= hy 


and, by the substitution of these values in the equations of 
the two conics, S and S' will become 


S= Aw?+2 Bay, + Cyr +2 Dixy +2 Lyyathz’ = 0, 
S! = Alas aL dew a. yA Fe = 0). 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 405 


The equation S+AS'=0 


will become therefore 
(A, + AA) ey? +2(B, + AB) MY +e = 0. 


In order that the conic S +AS'=0 be resolved into two 
straight lines, it is necessary and sufficient that the discrimi- 
nant of this polynomial in a, y,, 2, be zero, which gives to 
determine A a new equation of the third degree 


(7) Aj +- @,r + @',r? - AY = 0). 

This equation has the same roots as equation (1). For it fol- 
lows that both of the equations (1) and (7) give the values of A 
for which the equation S+.AS'=0 represents two straight 


lines. If equation (1) have three distinct roots, equation (7) 
will have the same roots, and consequently will | 


A,_ 9 _@)_ At, 

BO ON 
these relations will be identical if the coefficients A,, A‘, B,, 
B',, --» F, F"’; be replaced by their values as functions of A, 
A', B, B', --. F, F"; they will, therefore, still exist when one 
of the equations (1) or (7) have multiple roots. Therefore, in 
every case these two equations have the same roots. 


EXERCISES. 


1. Let M4, (a, y,) be a point taken on a conic whose equation 
is f(x, ¥) =0. Demonstrate: 
1° That the general equation of conics which osculate the 
conic f= 0 at the point WM, is 
S(@ y) FLA — %) + BY =) ] Pe, + Uf, +S.) = 9; 


dA and pu being two variable parameters. 
2° That the general equation of conics which are suboscu- 
latory to the conic f= 0 at the same point is 


Sa, y) + ALP, FW, +I F = 9, 


d being a variable parameter. 


406 PLANE GEOMETRY. BOOK III. 


2. The equilateral hyperbolas which osculate a given conic 
f=0 at a point M, pass through a fixed point P. Find: 

1° The locus of the center of these equilateral hyperbolas. 

2° The locus of the point P when the point M, describes 
the conic f= 0. 

3. Place two equal parabolas whose axes are perpendicular, 
so that they will osculate each other. 

4. Being given a parabola and a circle passing through its 
focus, find where the center of the circle should be in order 
that it have four real points of intersection with the parabola, 
or two real and two imaginary points, or four imaginary 
points. Study the form and the properties of the curve which 
separates these different regions (Ecole Polytechnique, 1865). 

5. Let 

S=au? +2buv + cv? +2 duw +2evw + fw? =0, 
S=alw+2 b’uv + cv? + 2 d'uw + 2 e'vw + f'w? = 0, 
be the tangential equations of two conics S and S'. It is 
required : 

1° To form the equation of the third degree which gives 
the values of uw in order that the equation 
: a+ pd’ = 0 
represent two points. 

2° Show in what way the roots of this equation in p are 
connected with the roots of the equation in A with respect 
to the two conics S and S’. 


RELATIONS CONNECTING THE ROOTS OF THE 
EQUATION IN 4. 
Let S = Av’? +2 Bry + Cy’? +2 Daz +2 Hyz + Fe =0, 

S'= Ale’? +2 Bleyt+ Cy’? +2 D'vz+ 2 E'yz+ F'? =0, 
be the equations of two conics which have been made homo- 
geneous by replacing # and y by _ and multiplying by 2’. 
The values of A in order that the equation 

S+dAS'=0 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 407 
represent two straight lines, are given by the equation of the 
third degree: 

(1) A+@0.4+ 0/74 AN = 0. 


We have seen above that the roots of this equation remain 
unchanged when the conics S and S' are referred to other axes. 
Suppose, more generally, that a, 8, y be three linear functions 
in a, y, 2, namely, 

a=av +by +c, 


(2) B=a'xe + d'y +c’, 
y= alas + bl'y + lz, 


which, equated to zero, represent three non-concurrent straight 
lines; one can find from these equations 2, y, z as linear homo- 
geneous functions of «, B, y, 


e=la +mB + ny, 
(3) yala +m'B +n'y, 
zg=l"atm'B+n"y; 


on substituting these values in the equations of the two conics, 
their equations will take the form 


S = Ay? +2 BeB + C8? +2DeB +24 By + hy =9, 
Ss = Aye + 2 B,'eB + O,'B? +2 D;'aB + 2 Hy 'By + f'y = 9, 
and the equation S + AS'= 0 will become 
(A, + AA,') 0? + 2(B, + AB,') aB +++ =0. 


In order that the conic S+AS'=0 become two straight 
lines, it is necessary and sufficient that this new homogeneous 
polynomial in «, 8, y decompose into two factors of the first 
degree, that is, that its discriminant 


(Ai AADC, + ACD +X + -- 


be zero; one has therefore, in order to determine A, a new 
equation of the third degree: 


(4) Ay + ©,A + ©, és + A,'A3 = 0. 


408 PLANE GEOMETRY. BOOK IIT. 


This equation has, moreover, the same roots as equation (1). 
Whence it follows from this result that each of equations 
(1) and (4) furnishes the values of A, for which the equation 
S + AS'= 0 represents two straight lines. If the first equation 
have three distinct roots, the second will have the same roots. 
Therefore, 

Beer Oye a i 
N16. Gear 


these relations will be identities if one replace the coefficients 
A,, A‘, B, B,', +» by their values as functions of .A, A’, 
B, B', +-. ete.; they exist moreover when equation (1) has 
multiple roots. Therefore, in every case, equations (1) and (4) 
have the same roots. 

We see finally how the roots of the equation in \ vary when 
all the coefficients of the equation S = 0 are multiplied by a 
constant factor A, and all of those of S'=0 by a constant 
factor A’. Then the equations of the two conics become 


KS =0, K'S'=0, 


and the general equation of the conic passing through their 
points of intersection becomes 


KS + r'K'S' = 0, 

an equation which is identical with S+AS'=0 if one put 
r! =n. Therefore, in order to obtain the values of X', for 
which the equation KS + r'K'S' = 0 represents two straight 
lines, it will be sufficient to take the three roots of equation 
(1) in A and multiply them by the factor + 

Summing up briefly, if the equations of two conics be trans- 
formed by substituting for a, y, 2 expressions such as (3) (that 
is, on referring them to any triangle of reference), and if the 
equations of the two conics be multiplied by constant factors, 
the roots of the equation in X remain the same or are multiplied 


by a constant factor. 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 409 


Whence it follows that: 

A homogeneous relation between the roots of the equation in X 
expresses a property of two conics independent of the choice of the 
co-ordinate axes, or, more generally, of the choice of linear func- 
tions a, B, y, that is, of the triangle of reference. Because a simi- 
lar relation exists when the axes or the functions «, 8, y have 
been chosen in a particular way, it will exist for every other 
system of axes or linear functions a, B, y. 

For example, if Aj, A», As be called the three roots of the 
equation in X, the relation A; — A, = 0 or, more symmetrically, 


Og ie 2, an 0, 


which is homogeneous and of the second degree in Aj, As, As, 
expresses the condition that the equation in » has a double 
root; that is, that the two conics are tangent. 

We proceed to determine the meaning of certain other 
simple relations. 

I. The relation dy + A, +A3= 0 or O' = 0 ts the necessary and 
sufficient condition in order that there exist a triangle inscribed 
in the conic S and conjugate with respect to S'. In case one such 
triangle exists, there exists an infinitude of such. , 

In order to prove this, suppose that there exists a triangle 
which is inscribed in S and conjugate to S’. Then, on calling 


a), By 
the equations of the sides of this triangle, the equation of the 
conic S will be - 
S=2 BoB +2 Dey +2 EBy = 0, 

and that of S' 

S'= Ale? + C'R? + Fy’. 

It is easily seen, on equating to zero the discriminant of the 
polynomial S + AS', that one obtains the equation in 


2 BDE — \(A'E’ + C'D? + F'B’) + 3 A'C!LF" = 0, 


in which the coefficient @' of A? is zero. Therefore, the condi- 
tion Ay + A, + Az = 0 is necessary. Conversely, if this condition 
be fulfilled in case of two conics S and S', of which the second 
‘S'is not resolvable into straight lines, there exists an infini- 


410 PLANE GEOMETRY. BOOK III. 


tude of triangles which are inscribed in S and conjugate to S'. 
In fact, take a point P on the conic S and construct the polar 
of this point with respect to S' (Fig. a); this polar intersects 





the conic S' in two points M and M' and the conic S at least 
in one point Q; let R be the harmonic conjugate of the point 
@ with respect to the two points M and M'; we shall show 
that this point R belongs also to the conic S. 

The triangle PQR being conjugate with respect to the conic 
S', the equation of this conic will be 


S'= Aleit Opa 2 0), 


on calling «= 0, B=0, y=0 the equations of the sides QR, 
RP, PQ of the triangle. The conic S passes through the 
point P the intersection of the sides B= 0, y=, and the 
point @ the intersection of the sides y = 0, «= 0; its equation 
will therefore have the form 


S= Fy’ +2 BoB +2 Day +2 EBy = 0. 


On forming the discriminant of the polynomial S + AS', the 
coefficient of A? will be : 


@'= ACT. 


Since this coefficient must be zero, and that neither A’ nor C" 
can be zero, because the conic S' would reduce to two straight 
lines, therefore will /=0, and the conic S is circumscribed 
about the triangle PQR conjugate to S'; what we wish to 
establish. 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 411 


II. The same relation 4; +A, + A3 = 0 or O©'=0 is the neces- 
sary and sufficient condition in order that there exist a triangle 
circumscribed about the conic S' and conjugate with respect to S. 
When one such triangle exists, there exists an infinitude. 

In fact, if there exist one such, and if a, B, y be called the 
first members of the equations of the sides of this triangle, the 
equation of S will be 


S= Ae?+ Cp’? + Fy’, 
and that of S' (§ 282. 2), 
S=pv+ fP+ry —2qrBy —2 rpya — 2 pqaB = 0. 


On forming the coefficient ©! of A? in the discriminant of 
S +S’, it will be readily seen that this coefficient is zero; the 
condition A, + A, + A; = 0 is, therefore, necessary. Conversely, 
suppose this condition fulfilled; select any tangent 77", a = 0, 
to the conic S' and its pole P’ with respect to S; from this 
pole two tangents P'T and P'T"' can be drawn to the conic S 
and one tangent at least P’Q' to the conic S'.. Let y=0 be 
the equation of this last straight line P’Q', and B = 0 the equa- 
tion of its conjugate P'2' with respect to the system of tan- 
gents P'T and P'T". Hence the triangle formed by the three 
straight lines «=0, B=0, y=0 will be conjugate to S, and 
two of its sides a=0, y=0 will be tangent to S'; we shall 
show that the third side B=0 is also tangent to S'. The 
equations of the two conics can be written: 


S = Ac’? + CB’? + Fy’ = 0, 
S'= pe? + PB? + 7°? —2qrBy + 2 Diay —2 pgoB = 0; 


since the first member of the equation of S' should reduce to a 
perfect square for «=0 and for y=0. The coefficient ©’ of 
d”? in the discriminant of S+AS' is C(p’r?— D”), and this 
coefficient should be zero. C cannot be zero, because if C were 
zero the conic S would reduce to two straight lines; one cannot 
have D' = pr, because for this value of D' one would have 


S'=(pa— 9B + ry)’, 


412 PLANE GEOMETRY. BOOK. III. 


and the conic S’ would be a double line. Therefore, D’= — pr, 
and the conic S‘ is inscribed in the triangle P’Q'R’ conjugate 
to S. 


RemMARK.—In case the conic S is circumscribed about a 
triangle conjugate to S', it is said, for brevity, that S is har- 
monically circumscribed about S'; then, according to what pre- 
cedes, the conic S' is also inscribed in a triangle conjugate to S, 
and it is said that S' is harmonically inscribed in S. 


ExAmpLe. —A triangle being given, there exists always a 
real or imaginary circle with respect to which the triangle is 
conjugate. For, on calling «=0, B=0, y=0 the equations 
of the sides of the triangle, the general equation of the conics 
conjugate to the triangle is 


Ad’? + CB? + Fv? = 0. 


The condition that this equation represents a circle gives two 
equations of the first degree, which determine the ratios of the 
coefficients A, C, F to any one of them. ‘The circle thus 
found is called the circle conjugate to the triangle; its center 
is the point of intersection of the altitudes of the triangle, 
because if from a point a perpendicular be dropped on the 
polar with respect to a circle, the perpendicular passes through 
the center of the circle. oe 

Having proven this, we proceed to demonstrate the following 
theorem : 

In case a conic is inscribed in a triangle, the power of the 
center of the conic with respect to the circle conjugate to the 
triangle is equal to the algebraic sum of the squares of the axes of 
the conic. 

Refer the conic to its axes, and let 


S' = Ae? +0" — A'C' =0 


be the equation of the conic, A’ and C" being the squares of 
the lengths of the axes. Let, moreover, 


S=e+y+2 Dxe+2 Ey+F=0 


Ni 


CHAP. XII SECANTS COMMON TO TWO CONICS. 4138 


be the equation of the circle conjugate to the triangle. Then 
the conic S’ will be inscribed in a triangle conjugate to S; 
therefore, if the discriminant of the polynomial S + AS" be 
formed, the coefficient ©’ of A? ought to be zero. This coeffi- 
cient is A'C’(F — A'— C"); since A’ and C’ are different from 
zero, we have - 
F=A'+0, 
which proves the theorem. 


Ill. The necessary and sufficient condition that there exist 
a triangle inscribed in a conic S and circumscribed about a 
conic S' is 

@”? — 4 @A'=0, 


or An +. x a. eg ae 2 AiA¢, amas 2 AoA See Z AsAy = 0, 
or Vai £ VA2 t Vg = 0; 


if there exist one triangle, there exists an infinitude. 


Let «=0, B=0, y=0 be the equations of the sides of a 
triangle inscribed in S and circumscribed about S'. The 
equations of the two conics can be written 


S=2 BaB+2 Dey+2 EBy = 0, 
S'= pre? + PB? + 1°? — 2 qrBy — 2 rpya — 2 pqaB = 0. 


If the discriminant of S + XS' be formed, it follows that the 
coefficients of A°, A’, A are 


A! pee ae 4 n°q?r", 
©' = 4 pqr (Ep + D¢ + Br), 
© = — (Ep + Dq + Br)’. 


It follows, therefore, that ©? —4@A'=0, a relation which 
can be written, owing to the relation between the coefficients 
and the roots, 


(Ay + Ag+ A3)” —4 (AyAz + AjA3 + sds) =a 0, 
or (5) AG + a + ae Een 2 Air ares 4 AvA3 reer v4 AsAy = 0. 


414 PLANE GEOMETRY. BOOK III. 


This last relation being homogeneous with respect to the roots 
of the equation in A, will hold, as has been seen, in whatever 
form the equations of the two conics may be written. It can 
be easily verified that this relation is equivalent to one of the 
following : 


VAL + VA2 EVA = 0. 


Conversely, if relation (5) hold, there exists an infinitude of 
triangles inscribed in S and circumscribed about S'.. This may 
be proven by following the method which has been employed 
in the cases of Propositions I. and IT. 

ExampLe I.— Consider two ellipses which have the same 
center and coincident axes, whose equations are 

a? y? Ss 
s=5+u- eee = ee a 
The values of A for which the equation S + AS'=0 repre- 
sents two straight lines are 
Ge oF 
A= 1, let As = Fa} 
therefore the necessary and sufficient condition in order that 
there exist a triangle inscribed in S and circumscribed about 
S' is 


ExaAmpLe II.— Consider two circles whose equations are 
Sf 7h 0, 8 = @ 0) fy 7 0; 
the coefficients of the equation in X are 
A'=7r, @=H+2r-—@, @©=2F'+r—-@; 


the necessary and sufficient condition in order that there exist 
a triangle inscribed in S§ and circumscribed about S’ is, there- 
fore, 
2°P+hR?—a@y—47°2R+r-d)=0 
or, simplifying, 
: (7? — Rk)? -47°R?=0, 


@?—-R=+2rhk, 


CHAP. XII. SECANTS COMMON TO TWO CONICS. 415 


a well-known relation connecting the radii of the circles and 
the distance between the centers of the two circles, the one 
circumscribed, the other inscribed or escribed to the triangle. 


EXERCISES. 


1. Show that the circle conjugate to a triangle is real when 
the triangle has an obtuse angle, and imaginary in case the 
angles of the triangle are acute. 

2. Prove that the locus of the centers of conics inscribed in 
a given triangle, so that the sum of the squares of their axes is 
constant, is a circle whose center is the point of intersection 
_ of the altitudes. 

3. In case a triangle is circumscribed about a parabola, the 
point of intersection of the altitudes is on the directrix. 

4. When a triangle is inscribed in an equilateral hyperbola, 
the point of intersection of the altitudes lies on the curve. 

(In these exercises, the point of intersection of the altitudes 
is regarded as the center of the circle conjugate to the tri- 
angle.) 

5. A parabola, y? = 2 px, 


and a circle, e+tyt2ax+2by+c=0, 


are given; determine the necessary and sufficient condition in 
order that there exist a triangle inscribed in the circle and 
circumscribed about the parabola. When will the circle pass 
through the focus of the parabola ? 

6. The equations of two conics being written in the form 


S= Ad?’ + BB? +C/ =0, 
S'= Ale? + B'B? + Cy =0 
what relation must exist among the coefficients : 
1° In order that S be harmonically circumscribed about S'? 
2° In order that S be circumscribed about a triangle cireum- 
scribed about S'? 


7. Two conics are tangent to each other at a point M; 
demonstrate that the necessary and_ sufficient condition, in 


416 PLANE GEOMETRY. BOOK III. 


order that there exist a triangle inscribed in S and circum- 
scribed about S', is that the radius of curvature of S' at the 
point M be equal to four times that of S at the same point. 

PARTICULAR CAsE.—If a circle which passes through the 
focus of a parabola be tangent to the parabola at a point M, the 
radius of curvature of the parabola at M is four times that of 
the circle. 

8. Consider a triangle and the circumscribed circle; there 
exists a conic tangent to three sides of the triangle and tangent 
to the circumscribed circle at a given point M. 

1° Find the center of the circle of curvature of this conic 
at M. 

2° Find the locus of this center when the point M describes 
the circumscribed circle. | 

9. What relation should exist between the roots of the 
equation in d in order that there exist a quadrilateral in- 
scribed in the conic S, and circumscribed about the conic S'? 

If there be one such quadrilateral, there will be an infinitude. 

As a particular case, we apply the relation found to the 
case where the conics are two circles. (See § 109.) 

10. Consider an ellipse #, of which the major axis and the 
focal distance are respectively 2a and 2c. Describe a circum- 
ference of a circle C with the radius V2 (a? + c’) about one of 
the foci F of the ellipse as center. A tangent P,P, is drawn 
from any point P, of the circumference C to the ellipse; from 
the point P,, where it intersects the circumference (/ again, a 
second tangent P,P; is drawn to the ellipse ; finally, from the 
point P,, where this second tangent intersects the circumfer- 
ence C, a third tangent P,P, is drawn to the ellipse which 
intersects the circumference in the point P, It is required 
to prove that the second of the tangents drawn from the point 
P, to the ellipse passes through the initial point P,. (Ecole 
Normale, 1885.) 


CHAP. XII. PROPERTIES OF POLYNOMIALS, ETC. 417 


THE APPLICATION OF THE PROPERTIES OF HOMOGENEOUS 
POLYNOMIALS TO THE THEORY OF CURVES OF THE 
SECOND DEGREE. 


Let f(a, y, 2)= Ax’ + 2 Bey + Cy? + 2 Daz + 2 Hyz + Fe 


be a homogeneous polynomial of the second degree in a, y, z. 
It is known that if the discriminant A be different from zero, 
the polynomial f is resolvable into a sum of three squares 
linearly independent; if this discriminant be zero without all 
the minors being zero, the polynomial can be decomposed into 
a sum of two squares linearly independent; finally, if all the 
minors of the discriminant are zero, the polynomial is a per- 
fect square; the converse statements are true. 
1° Suppose that the discriminant 


A= ACF — AE’ — CD? — FB? +2 BDE 


is zero, and similarly all of its minors a, b, c, d,e,f. Then 
the polynomial f is the square of a linear function 


S (a, y, z)= a (la+ my + nz)’, 


a being a constant which is positive or negative; and on 
representing the function la+my-+nz by a, it follows that 
one has identically /',=2ala, f',=2ama, f',=2ane. In 
this case, if w, y, and z be regarded as the homogeneous co-ordi- 
nates of a point, the equation 


Se, y, z= 0 
represents two straight lines coincident with the straight line 
lx + my + nz = 0, and the three equations 


f2=9, fy =9, fi. =9 
represent this same straight line or are identities, since, for 
example, the equation f', = 0 when n= 0. 
2° Suppose that the discriminant A is zero and that its 


minors. are not all zero. Then the polynomial f can be 
resolved into a sum of two squares linearly independent, 


(1) S(@, Y, 2)= ac’ + bp”, 
2D 


418 PLANE GEOMETRY. BOOK IIT. 


where a and 0 are constants and « and £ are linear homo- 
geneous functions of a, y, z: 


a=le+myt+nz, B=l'x 4+ m'y + n'z. 


To say that these functions are linearly independent, is to say 
that there do not exist two constant factors k and k', both of 
which are not zero, and such that ka + k'B is identically zero. 
The polynomial f may be written in an infinite number of 
different ways in form (1); we shall show how all of them 
can be obtained. The identity (1) can be written 


f(@ y 2)=(aVat BV— 0) (eVa—Bv—D,, 
or S (a, Y; 3 PQ, 


where P and @ designate two homogeneous linearly independ- 
ent functions in a, y,z. These two linear functions are easily 
found; in fact, if the three coefficients A, C, F are not all 
zero, the polynomial f will be a trinomial of the second degree 
in a, y, or 2, and this trinomial can be resolved into factors 
of the first degree, which will be Pand Q; if A, C, and F be 
zero, the discriminant reduces to 2 BDE, and since it is zero, 
one at least of the three coefficients B, D, E is also zero, 
and then one of the three variables a, y, or zis a factor, and 
the decomposition is immediate. The polynomial being thus 
put under the form PQ, all possible decompositions will take 
the form of a sum of two squares, on noticing that one has 
identically | ef 


(2) f(a, 4 2=PQ= i [AP + 1Q)?—(AP— 1)", 


where A and p designate constant coefficients different from 
zero. On allowing A and p» to vary, there will be an infinitude 
of decompositions of f into two squares: one has all of them, 
because if one imagines any decomposition 


Sf = ae) + 0,8" = (uyV a + Biv — b;) (Va, — Biv —d,), 


in which a, f, are linear functions, a, and b, constants, one 
would have identically 


(c-V ay + Biv — b;) (0 V ay — piv = b)= PQ, 


CHAP. XII. PROPERTIES OF POLYNOMIALS, ETC. 419 


whence, on designating a constant by k, 
Vay = Biv — b,=kP, 


oy Vay ai Biv — by = © 


ty Va, = (P+ =) Bv—t=3(P— 2), 
and finally 


foawtt npin7 (er + 2) — (aP— a 


an expression which becomes identical with (2) on supposing 


Ah, ° 
The values of a, y, z, 


C= UM, YHYW %*=— 424, 


which reduce simultaneously to zero the linear functions P and 
Q, and, consequently, the functions « and 8, which are equal to 
AP + pQ and AP — wQ, reduce to zero the three partial deriva- 
tives f' fy fe 

If the coefficients of the polynomial f be real, the factors P 
and @ can be real or imaginary. In order to obtain, in the 
decomposition of (2), the squares in case of real coefficients, it 
will be necessary, if P and Q be real, to take \ and yp real; 
then one of the squares which appears in formula (2) has a 
negative coefficient, the other a positive; if Pand Q be imagi- 
nary, one could put, since their product is real, 


P=pt+iiq Q=h(p—i9), 


where h represents a real constant and p and q are real linear 
functions; one puts then A=A(A'+ ip’), w=A'—ip', and it fol- 
lows: 


f= —_ _[a'p— n'a) 409+ wp)? 1, 


r” fn 
where the two squares have the same sign as that of h. 

The geometric interpretation of these results is very simple. 
On considering 2, y, z as the homogeneous co-ordinates of a 


420 PLANE GEOMETRY. BOOK III. 


point, it, follows that the identity f(a, y, z)= PQ shows that 
the equation f= 0 represents two distinct straight lines, real or 
imaginary. If z, be different from zero, the two straight lines 
aX nh, : 


b 
2 ey 


intersect in the point whose Cartesian co-ordinates are 


the straight lines 
«=P +pQ=0, B=AP—pQ=0 


pass through this point and are harmonic conjugates with 

respect to the two straight lines P=0 and Q=0. If 2, be 

zero Without either of the functions P and Q reducing to the 

form nz, the two straight lines are parallel and have as common 
a 

angular coefficient _ The straight lines «=0, B=0 are 


1 
parallel to the same direction and are harmonic conjugates 
with respect to the straight lines P= 0, @=0; one has iden- 
tically, in the present case, 


P=mQ+ nz, 


mand n being constants; if therefore one put mA =— p, the 
straight line « = (0 becomes the straight line at infinity and its 
conjugate B=0 becomes the straight line equi-distant’ from 
the parallel straight lines P= 0, Y@=0: 


2mQ + nz= 0 
since f(a, y, 2) = PQ = (mQ+ nz) Q, the two equations 
fic = Q,(2 mQ + nz) = 0, 
St, = U,(2mQ@ 4+ nz) =0 


represent this same straight line, provided that neither of them 
be identically zero, which would happen, for example, if the 
quantity @!, were zero. 


ft 
ve 


Finally, if one of the two functions P or Q, Q for example, 
were of the form nz, the straight line Q = 0 would be removed 
to infinity, the equation f = 0 would represent a single straight 
line P=O at a finite distance, and the two straight lines 
a=0 and B=0 would be parallel to this straight line and 
situated at equal distances on either side of it. 


CHAP. XII. PROPERTIES OF POLYNOMIALS, ETC. 421 


3° Suppose, finally, that the discriminant A be different 
from zero. In this case, the polynomial f(a, y, z) can be 
decomposed into three linear independent squares, 


(3) S (a, yy 2) = ae + DB’ + cy’, 


where a, b, c represent constants different from zero and «, B, y 
linear functions : 


a=let+my+nz, B=lVe+m'y + n'z, y= uxt vy + U%, 


such that the determinant 


eon on 4 
l' m' m!' 
1 oO 


be different from zero. 
In order to obtain all the decompositions of f into three 
squares of the form (3), we notice that one of the three linear 
functions a, B, y can be chosen arbitrarily, the function y for 
example, with the condition that the coefficients u, v, w of this 
function DO NOT REDUCE the following polynomial to zero, 


4 U,V, W) = au? + 2 buv + cv? + 2 duw + 2 evw + fw’. 
p 


In fact, u,v, w being chosen arbitrarily, let us consider the 


difference 
F(a, Y; Z)= f(a, Y; oe ry’; 


where X is a constant; this difference F’ is a homogeneous poly- 
nomial of the second degree in a, y,z. Determine Xd so that the 
discriminant of F(a, y,z) be zero; we will have the equation 
A—dwW B-—dAuw D—duw 
(5) B—duw C—-dr’ E—)dvw |=), 
D—d\uw E—-dw F-dA,2w 
whose development with respect to powers of A is obtained by 
putting in the equation in A in § 286 


Fi=—v’, 


422 PLANE GEOMETRY. BOOK IIT. 


and consequently A'= 0, 6'= 0, © =— ¢(u, v, w), d represent- 
ing polynomial (4). Equation (5) reduces, therefore, to the 
equation of the jirst degree 


A — rAP(u, v, w)= 0, 


which determines d if d(u, v, w) be not zero. Letcbe the value 
of A deduced from this equation: 


A 
¢e= 
p (U, 2, w) 


The discriminant of the function F = f — cy? being zero, this 
function can be decomposed into a sum of two squares; it can- 
not be a perfect square aa’, because if it were such one would 
have | 

SY, 2) = Cy? + ae? ; 


the function f would be a sum of two squares and its discrimi- 
nant A would be zero, which contradicts the hypothesis. The 
polynomial F’ = f— cy’ being decomposable into a sum of two 
squares, one can apply to it what has been said in the preceding 
paragraph and find all possible ways of putting it in the form 
au’ + 68°. To each of these decompositions of F (a, y, z) into 
two squares will correspond one decomposition of f into three 
squares 

SF = an’ + dB? + cy’, 
' y having been chosen arbitrarily. 
If F be decomposed into two factors 


(6) FQ, y, 2)=f— cy’ = PQ, 
the values x = 2, y= y,, 2 = 2,, Which reduce P and Q to zero, 
reduce « and 8 to zero, and reduce also the partial derivation 
reat, tt, Ft . 00 Zero. 

Since 

}PL=h/.—cuy, $F) =4f,—cvy, 4 P= hss — wy, 
one has, on replacing 2, y, z2 by %, y, %, and representing the 
quantity ua, + vy, + wz, by y, 


(7) + S's, = Uy1, Ree = Vy a's, = CWy1 5 


CHAP. XI. PROPERTIES OF POLYNOMIALS, ETC. 428 


the constant y; is not zero, because, if it were, f’,, Py, J, would 
all be zero, and the discriminant A would vanish. One has 
identically 


(8) 3 (af, a WT, ake “f"z,) = oy ya 
and, on putting, in this identity, # = a, y = yy, 2 = %, and apply- 
ing the theorem of homogeneous functions, 
S (@1y Yay 2) = Cyr’ 
Whence it follows that 
Gf, + ul ,+o.)° 
4 f (2%, Ny z) 


and, on replacing cy? in the identity (6) by this expression, one 
gets, after removing the denominator 4 f(a, %, %1), 


(9) Af (2, yy 2)F (By Yr» 2) — (F's, + Uy, +e)” 
=4 PQS (ey Yiy 21): 


Finally, if in the relation which determines ¢, 





aes 2 
= Cy, 


ch (u, Vv, wW)=A, 


u, v, w be replaced by their values deduced from relations (7), 


ne 1 Jy, w= 1 EN 
2 cy,’ PD Cy,’ 








ut 
Us 9 
it follows, since the polynomial ¢ is homogeneous, 
ee 4 
6 (SS 29 $F t's, = As 
Cy 


and, on replacing cy,” by its value f(a, %, %); 


(10) d's.) St yp 2 ele J= Af (at, Yay %)5 


which gives a remarkable identity. 


Gromerric INTERPRETATION. — On considering 2, y, z as the 
homogeneous co-ordinates of a point, it is plain that the equa- 


424 PLANE GEOMETRY. BOOK III. 


tion f(#, y, z)=0 represents a conic not reducible to two 
straight lines, real or imaginary ellipse, hyperbola, parabola. 
The identity 

S(&, Y, 2) = ac? + OB? + cy? 


represents the first member of the equation of this conic de- 
composed into a sum of three squares. If the three coefficients 
a, b, c have the same signs, the curve is an imaginary ellipse ; 
if not, it is a real conic: it is known, moreover, that in all 
possible decompositions of f into three squares, the number of 
coefficients a, b, ¢ which have a definite sign is invariable. One 
of the functions «, B, y can be chosen arbitrarily, for example 
the function y= wa + vy + wz, with the condition that one 
does not have $(u, v, w)= 0, that is, with the condition that 
the straight line y=0 is not tangent to the conic (§ 126). 
Suppose that this condition is fulfilled, the equation of the 
conic could be written in the form 


F(a, Y; z)= cy’ + PQ=0, 


which shows that P= 0, Q=0 are the equations of the tan- 
gents at the points where the straight line y = 0 intersects the 
conic. We have called a, y, 2, the values of a, y, 2 which 
reduce P and @ to zero, that is, the homogeneous co-ordinates’ 
of the point of intersection of the two straight lines P= 0, 
@=0; the straight line y=0 is the chord of contact of the 
tangents emanating from this point, or the polar,of this point; 
owing to the identity (8), the equation of this straight line can 
be written 


af, tur, +2, =0; 


which is the well-known equation of the polar of the point 
with the co-ordinates a, 7, z. Owing.to the identity (9), the 
equation PQ=0, which represents the ensemble of the tan- 
gents drawn from the point (a, 4, 2) to the eonic, can be 
- written 


AP (2, Y, 2S (Ay Ivy %)—(f',, + uf, +f.) = 0. 


Finally, identity (10) shows that the necessary and sufficient 
condition in order that o(f ‘ny J yy J'2,) be zero is that f(a, %, 21) 


CHAP. XII. PROPERTIES OF POLYNOMIALS, ETC. 425 


be zero and conversely, which means geometrically that the 
necessary and sufficient condition in order that the polar of 
the point 2, y;, 2, be tangent to the curve is that this point be 
on the curve. ‘ 

The straight lines c=0, B=0 pass through the point of 
intersection of the tangents P=0, Q@=0, and are harmonic 
conjugates with respect to these tangents. If z, be different 
from zero, the two straight lines P= 0, Q=0 are concurrent 
in a point situated at a finite distance, whose Cartesian co- 


ordinates are —, If z,=0, these two straight lines are 


1 %4 
parallel with angular coefficient = or one of them is at infinity, 


£ 
which happens when one of the functions P or Q reduces to 
the form nz; in this case (z,=0), the straight line y=0 has 


the equation 
Uf, + WI, = 9; 


it coincides with the conjugate diameter of the direction of the 
two straight lines P=0, Q=0, or of that of the two which 
is at a finite distance; whence it is said that y = 0 is the polar 
of the point at infinity (@, 4%, 0). 
Thus, if the equation be written in the form 
ST (a, y, 2) = aa? + v6? + cy =0, 
the straight line y = 0 is the polar of the point of intersection 
of the other two straight lines c=0, 8 =0; since the same is 
true in regard to the straight lines « = 0, B = 0, it follows that 
the triangle formed by the three straight lines a=0, B=0, 
y = 0, is a conjugate triangle with respect to the conic. 
Application to the reduction of the equation of the second de- 
gree. 
1. Assume that f be different from zero; then take y =z, 
that is, w= 0, v= 0, w =1, and we obtain 
peer need 
$(0,0,1) 
Identity (6) gives, in this case, 


S (&, Y, 2) — 2 = PQ = aa? + bp. 


426 PLANE GEOMETRY. BOOK III. 


The co-ordinates of the point of intersection of the straight 
lines P= 0, Q = 0, satisfy the equations 


7 =, F,=%, 


which follows from relations (7), where one supposes u=v=0; 
this point is the center of the curve. The straight lines 
P=0, Q@=0 are the asymptotes: their homogeneous eqiation 
is therefore | 


A 
Sf (® Y 2) eee 0, 
or, in Cartesian co-ordinates, 


F(X, ¥,1)—2=0. 


These asymptotes, P = 0, Q = 0, can be real or imaginary. In 
the first case the curve is a hyperbola, the coefficients a and b 
have opposite signs; in the second case it is an ellipse, the co- 
efficients a and b have the same signs. The straight lines 
a= 0, B =0, whose equations have the form 


AP + pQ = 0, AP — pQ=9, 


are harmonic conjugates with respect to the asymptotes; they 


are two conjugate diameters; if the ratio A be so determined 


A 
that these straight lines are 5 Sahitya each other, they 
coincide with the axes. On taking the straight lines a = 0, 
8=0as axes of Cartesian co-ordinates, the equation of the 
conic takes the simplified form, 
TD. Goa bY? 42 (). 

2. Suppose f=0. Then the preceding method of reduction 
is no longer applicable, because (0, 0, 1)=0. We give a 
second method of reduction which is applicable in all cases. 
It has been proven that if x, 4, % be the homogeneous co- 
ordinates of any point not situated on a conic, 


A f(a, Y; z) f (a, Yi Z) — (af, a5 Cle Lm Ae 2 PQ, 


where P= 0, Q@=0 are the equations of the tangents drawn 


CHAP. XII. PROPERTIES OF POLYNOMIALS, ETC. 427 


from the point (a, 4%, 2) to the curve. Take, in particular, © 
z, = 0 with the condition f(a, y, 0) 2 0, and notice that 


af, + i + af", i= tf', + WS'y + ZF es 
we have 


Af (a, y, 2) f (ay Yr 0)= (af, + nS)? + PQ; 


the straight lines P=0, Q=0 intersecting at the point at 
infinity (a, 4,0), are parallel, or one of them is at infinity. 
One will have, for example, 


Q=mP + nz, 


where m is zero, if Q be at infinity. Then the equation f= 0 
can be written 
(af, +f", + mP? + nPz = 0. 


On putting z = 1, one will obtain the equation of the curve in 
Cartesian co-ordinates; then one chooses the straight line 


af, + ntl = 9 
for the new axis O'X', and the straight line P = 0 for the axis 
O'Y’, and the equation will take the reduced form 
Veo gx"? = 0; 


in the particular case when the straight line Q=0 is at in- 
finity, one has m = 0), therefore g = 0, and the equation takes 
the very simple form 
F y” eo 2 pxX'! — 0, 
which represents parabolas. 

The ratio “ could be so determined that the straight line 


vy 
tf", + nS y= 0 


is perpendicular to P=0; the first straight line will then be 
an axis of the conic and the second the tangent at the vertex. 


Book IV* 
THE GENERAL THEORY OF CURVES 


——-ot@too— 


CHAPTER -I 


THE CONSTRUCTION OF CURVES IN RECTILINEAR 
CO-ORDINATES. 


334. The construction of a curve is simply: the graphic 
representation of the trace of the real function of a single 
variable, when this variable is allowed to change in a con- 
tinuous manner. If the values of y which correspond to the 
various values of w be calculated, a certain number of the 
points of the curve can be constructed, but these points are not 
sufficient, even for an approximate trace of the curve, because 
they can be connected in various different ways, and, more- 
over, it can happen that, between two ordinates which are very 
nearly equal, the curve has infinite branches. It is, therefore, 
indispensable first of all to know by some general method the 
trace of the function which represents the variations. 

When the equation is solved with respect to one of the vari- 
ables, y for example, one considers each of the determinations 
of y in particular, and examines them for the limits of « for 
which y remains real. Let x and 2, be the two limits; if the 
value of y remain finite in this interval, it furnishes a finite 
branch of the curve; if the value of y becomes infinite for one 
or more intermediate values a, 0, -+- of the variable, one has 
various infinite branches, asymptotic to the straight lines that 
correspond to the values of x, which make y infinite; in such 
a case the interval x to 2, is subdivided into several intervals: 

428 


CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 429 


the first from x to a, etc., in such a way that in each of them 
the ordinate does not become infinite. Afterwards one ex- 
amines how -y varies in each of the intervals, for example as x 
increases from a) to a. Sometimes one perceives immediately, 
from the expression for y, how this quantity varies, but more 
often that is not the case; in this case, however, one has recourse 
to the derivative. It is known, moreover, that if the function 
remains finite, as the variable x increases from a certain value, 
the function will vary in the same sense as the variable so long 
as the derivative preserves the same sign; the function in- 
creases if its derivative be positive and decreases if its deriva- 
tive be negative. Let «, 8, y, -+- be the successive values of x 
comprised between a) and a for which the derivative changes 
in sign. As the variable x increases from 2 to @, the deriva- 
tive preserves the same sign, for example the sign +, and the 
function increases; from « to B, the derivative is negative and 
the function decreases, etc. We have demonstrated that the 
angular coefficient of the tangent at any point of the curve is 
equal to the value of the derivative at this point. Thus, the 
sense in which the ordinate of the curve varies is indicated by 
the angular coefficient of the tangent. 

When the derivative changes its sign from positive to nega- 
tive, the ordinate ceases to increase and then decreases ; it 
attains therefore a maximum value. If, on the contrary, the 
derivative change from negative to positive, the ordinate ceases 
to decrease and then increases; it attains therefore a minimum 
value. It should be noticed that these terms maximum and 
minimum should not be taken with their literal meaning; they 
indicate only the comparison of a particular value of the 
ordinate with its neighboring ordinates. 

In general, the derivative, remaining finite and continuous, 
changes in sign on becoming zero, and consequently the tan- 
gents at the points whose ordinates have the maxima and 
minima values are parallel to the axis OX. Every value of 
«2 which makes the derivative zero does not necessarily give 
a maximum or minimum value of the ordinate; one must 
examine if the derivative change in sign: moreover, in all the 
cases, the tangent is parallel to the axis OX. 7 


430 PLANE GEOMETRY. BOOK Iv. 


335. Exampre I. — The strophoid defined in § 23 has the equation 





a@—2 


=+2 
y=t are 





When « varies from zero to — a, the numerical value of y increases 
continually from zero to infinity ; whence one obtains the two infinite 
branches ON, ON’ asymptotic to the straight line HH’ (Fig. 18). If x 
vary from zero to a, the ordinate y begins with the value zero and returns 
to zero, passing always through finite values; it begins therefore by 
increasing, then later it decreases, and consequently it passes through a 
maximum value ; but one does not see if the function does not experience 
in the interval several alternatives of increasing and decreasing. The 
positive value of y has the derivative 


, —%@%—ar+ a? 


: Vat x)(a—2) 








The numerator becomes zero for two values of x, the one x, positive 
and less than a, the other negative. When x varies from zero to 2, the 
derivative is positive, the function increases ; from 2; to a, the derivative 
is negative, the function decreases ; the ordinate is a maximum for the 
value . 
v5 —1 

2 





% =a ’ 


equal to the greater segment of the line a divided in a mean and extreme 
ratio. 


336. The tangent can often be determined at certain points 
of the curve, or, what amounts to the same thing, certain par- 
ticular values of the derivative, without recourse to the general 
expression for this derivative. Consider, for example, the 
point O of the strophoid; join this point to a neighboring 
point M whose co-ordinates are w and y; the angular coefficient 


; 
of the secant OM is equal to the ratio : ; the angular coefficient 


at the point O will be found on seeking the limit of this ratio 
as « approaches zero. Here one has 


' Ae — 2, 

wo = ato’ 
when z approaches zero, this ratio has the limit +1. The two 
branches which pass through the point O have as tangents at 





CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 481 


this point the bisectors of the angles of the axes. The tangent 
at the point A would be found by considering the ratio -—— ee} 
this ratio increasing indefinitely as x approaches a, the caneeat 
at the point a is parallel to the axis Oy. 


337. Exampte II. — We propose to study the curves represented by 
the equation y? = Ax + Bu? + Cx + D (it can be demonstrated that these 
curves, reproduced by projection, represent all the curves of the third 
degree). One can assume that the coefficient A is positive without chang- 
ing the direction of the x-axis. There are several cases to consider : 
1° The three roots of the polynomial of the third degree are real and 
unequal; let a, b, c be these roots arranged in order of increasing magni- 
tude ; then we may write 


= A(x — a)(x% — 6) (a —¢). 


The ordinate is imaginary when « varies from — o to a; real when x 
varies from a to b; imaginary when & varies from b to c; real when « 


Fig. 206. Fig. 207. 


as = 
ee 


Fig. 208. Fig. 209. Fig. 210. 

















varies from c to +o. ‘The curve is composed of a closed oval and an 
infinite branch (Fig. 206). 2° When the two roots a and b become 
equal, the oval reduces to a point a (Fig. 207). 38° When the two roots 
band c are equal, the oval becomes united to the infinite branch at }- 
(Fig. 208). 4° If the three roots a, b, c are equal, the curve has a cusp 


432 PLANE GEOMETRY. BOOK IV. 


at a (Fig. 209). 5° Finally, if the polynomial of the third degree has but 
one real root a, the curve has the form given in Fig. 210. 
The angular coefficient of the tangent is given by the formula 


»__ 3 Ax? + 2 Bet+ C _ 3A? +2 Br+ C 
2V Ax? + Bu? + Cx +D 2y 











In the first case, the numerator, which is the derivative of the polynomial 
of the third degree, becomes zero for a value a! comprised between a and 
b, and for a value b’ comprised between } and ¢; to the first corresponds 
the maximum value of the ordinate in the oval. In the third case, the 
numerator becomes zero for the double root b ; the denominator becoming 


zero also, the formula assumes the indeterminate form ~ and no longer 
determines the tangents at the double point b; they may be found by 
determining the limit V.A(b — a) of the ratio 





. = 5 as x approaches b. 
338. When the equation, supposed algebraic, is not solved, 
whether this solution is possible or not, or whether it is 
deemed useless to solve it, we can often, by employing the 
theorems concerning the roots of equations, construct the 
curve. | . 
Certain properties of the curve can be immediately recog- 
nized by inspection of its equation. 1° When the equation 
has terms all of which are of evén degrees, or of odd degrees, 
it is clear that, if it be satisfied by a=«a, y=, it will also 
be satisfied by e=—a, y=—f; that is, the two points 
(«, B), (— a, — B) are situated symmetrically with respect to 
the origin; therefore this point is the center of the curve. 
2° If the equation contains only even powers of one of the 
variables, y for example, the real values of y, which correspond 
to a particular value of a, are two by two equal and of contrary 
signs; if the axis be rectangular, it follows that the points of 
the locus are situated symmetrically with respect to the a-axis, 
which is an axis of the curve. 3° When the equation of the 
curve remains unaltered, when a is changed into y and y into @, 
if the equation be satisfied by «=a, y=, it will also be 
satisfied by «= 8, y=«; the two corresponding points are 
situated symmetrically with respect to the bisector of the angle 
YOX, which is an axis of the curves. It follows similarly 


CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 483 


that, if the equation does not change by substituting — y for 
2 and — x for y, the bisectors of the angle YOX' is an axis. 

Let f(x, vy) =0 be the equation of the curve; it is known 
_ Sle, y) 

(2 9) 
The expression for y' contains the two variables # and y; it 
gives the angular coefficient of the tangent at every point 
whose co-ordinates are known, excepting at the points where 
the two partial derivatives are zero at the same time. 


that the derivative y' is given by the formula y’= 


339. Examp ce III. — Construct the locus of the points such that the 
product of their distances from two fixed points F and F' is equal to a 
given number. 

Take as origin the mid-point O of the straight line FF’, this straight 
line as the x-axis, and a perpendicular to it as the y-axis; let 2c be the 
distance FF’, a? the constant product, the equation of the locus is 


(1) y* + 2 (a2 + 07) y2 + (a? — c?)2? -— at = 0. 


This equation involves only the even powers of each variable; each 
axis is therefore an axis of symmetry of the curve, and the origin is at 
the center. On considering y? as unknown, equation (1) is of the second 
degree ; the binomial B? — 4 AC reduces in this case to 4 (4 c?x? + at), a 
quantity which is always positive: the roots are therefore always real. 
When the last term (x? — c?)?— a‘ is positive, the values of y? have the 
same sign, and since their sum — 2 (#? + c?) is negative, the two values 
of y? are negative and the four values of y are imaginary. In order that 
equation (1) has real roots, it is therefore necessary that we have 


(a? — c?)2?— at <0, or (4? — c? — a?) (2? — c2 + a?) <0, 


and, consequently, 
w2<a2+ c? and «2 >c? — a. 


Then one of the values of y? is positive, the other negative. 

Take OA = OA! = Va? + c?; the curve lies between the straight lines 
drawn through the point A and A’ parallel to the y-axis. The second 
condition gives rise to the discussion of several cases. 

1° a<c. Take OB=OB' = Vc? — a2, and draw at the points B and 
B' lines parallel to O Y (Fig. 211). The curve consists of two parts, one of 
which is comprised between the parallel lines drawn through the points 
Band A, and the other between the parallels drawn through the points B! 
and A’. If x be given one of the values OB or OA, one of the corre- 
sponding values of y? is zero, and the other is negative; as x increases 


25 


434 PLANE GEOMETRY. BOOK IV. 


from OB to OA the value of y?, which at first is zero, increases, then de- 
creases and becomes zero again; we obtain thus a closed curve BOAD. 





’ 

! 
——-t 
| 

' 

! 

| 
ate 
! 

! 

| 

| 

! 

| 

' 

' . 

' 

! 
ioe er 
! 

i] 

| 

‘ 
) ‘ 





’ 
‘ 
' 
1 
\ iad 
' 
y 
ea” 
° 
& 
s—----=s 
' 
1 
eh. Lee 
ae ear 
a 


Fie Fo 





The negative values of x give a second curve B’C’A'D , equal to the pre- 
ceding. . 
The angular coefficient of the tangent is determined by the formula 


et eo) 
2 ! = 
( ) y y (a a y? + c?) 





At the points A and B, y is zero and y/ is infinite; the tangent is there- 
fore parallel to the y-axis. The numerator of y! becomes Zero when 
a2 + y2=c%, From the point O as center, describe a circle with OF as 
radius. The circle intersects the curve in four points C, D, C’, D’, given 
by the formulas ; 
4ct—at ._ a i 


2 — ica 
40° 4 


Since the arc BC lies within the circle, at any of the points of this arc, 
the function #2 + y? — c2 has a negative value, and y! is positive. For 
points of the arc C’A, the factor «? + y? — c? is positive, and y' is negative. 
Hence from B to C the ordinate increases, and from C to A it decreases ; 
the ordinate at the point C is a maximum. 

2° a=ec. The second condition is satisfied whatever x may be; x 
may vary from —cV2 to cV2. When « varies from 0 to cV2, the posi- 
tive value of y? begins with zero, increases, then decreases and becomes 
zero again; we have a closed curve OCADO (Fig. 212), which passes 
through the origin: to the negative values of « there corresponds a curve 
which is the symétrique of the preceding with respect to the y-axis. The 
circle of radius OF intersects the curve in four points, whose co-ordinates 


CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 435 


have a numerical value 7 which is a maximum ; the abscissas of these 


points have an ab- 


cv3 


solute value oN 


This curve is called 
the lemniscate. 

At the origin the 
value y' takes the 
form 5 it is easy 


to show that this 





is the case at the 
multiple point of 
any algebraic 
curve. In fact, the 
value of y’ is given 
by the formula 


_ S(t, Y), 


a S'y(% Y) 





Y 





Fig. 212. 


Since f(x, y) is an integral polynomial with respect to « and y, the 
partial derivatives f’,(x, y), f',(a, y) are also integral polynomials with 


respect to the same variables. If 
these polynomials do not become 
zero, when x and y are replaced 
by the co-ordinates of the multi- 
ple point, y’ will have at this point 
a unique value, whereas it should 
have as many different values as 
there are branches of the curve 
which pass through the multiple 
point. In the present case, the 
equation being a bi-quadratic can 


be solved with respect to 7; to 


each value of y there corresponds 
a derivative which has a definite 


value when x is put equal to zero. 











Fig. 213. 


This value of the derivative is, as 


has been noticed in § 336, the limit of the ratio _ when « approaches 


zero. The limit of this ratio can be found without solving the equation. 


’ Put a= t, or y= tx; on substituting in equation (1), it becomes 


ax7t* + 2 (4? + c2) 2 4+ a2 2c? = 0. 


When « is very small, one of the values of ¢? is approximately equal to 
unity, the other is negative and very large in absolute value ; on confining 


436 PLANE GEOMETRY. BOOK IV. 


ourselves to the real values of y, we have lim , =+1. The tangents at 


the point O bisect the angles formed by the axes. 


3° a>c. The second condition is satisfied, whatever « may be; x 


can therefore vary from 


—-Ve+a@ to +Ve4+ a. 
For x equal zero, the positive value of y? is @2 — c2. Take on the y-axis 
OB = OB! = Va? — ¢’, 


the curve passes through the points B and B’. If x vary from 0 to 

c2 + q?, y? begins with a? — c?, decreases, and becomes zero ; the locus is 
a closed curve whose vertices are 
the points A, A’, B, B’. In order 
that the circle intersect the curve, 
it is necessary that a<cv2. 
When this condition is satisfied, 
the ordinate increases from B to 
Cand diminishes from C to A: 
the ordinate of the point B is a 
minimum, that of Ca maximum. 
If, on the other hand, one have 
a>cv2, the circle lies within 
the curve, of which the ordinate 
diminishes from B to <A; the 
ordinate of the point Bisa maximum, In Fig. 214 it is supposed that A 
is equal to eV 2: 











Fig. 214. 


340. Exampte IV. —Construct the curve 


(1) 2y° — 5ay?+ x5 = 0. 


ft 


This equation being of the fifth degree with respect to each of the 
variables, it cannot be solved with respect to either of the variables ; it 
involves only terms of odd degrees ; therefore the origin is at the center 
of the curve. Examine how many of the roots of the equation, in which 
y is regarded as unknown, are real for various values of x. 

Suppose in the first place that « is positive, equation (1) will have at 
most two real positive roots, since its first member has but two variations 
in signs. The derivative of the first member with respect to y is 
10y(y3 — x). This derivative is negative from y = 0 to y = Vx, positive 
from this value to infinity. The first member, which is positive for y = 0, 
decreases when x varies from 0 to Vz, and increases indefinitely as y 
becomes greater. The equation has therefore two positive roots, or it does 
not, according as the value y = V« renders the first member negative or 
positive, that is, according as one has #9 < 27, or 2!>27. If y be 


CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 487 


changed into — y, the first member has but one variation ; therefore the 
equation has one negative root and only one. 

For «= 0, the five roots of equation (1) are zero; for values of x 
between 0 and 27, the equation has two positive roots, and one 
negative ; for x — 27, the two positive roots are equal, because they 
reduce the derivative to zero. When « becomes larger than V27, the 
equation has but one real root, which is negative. The two positive roots 
give an oval OABO (Fig. 215), comprised within the angle YOX, and 
the negative root a finite branch OC situated in the angle Y’OX. To 
negative values of «x there corresponds y 
an oval OA'B’O and infinite branch 


OC', the symétrique of the preceding Be = 

with respect to the center. ‘The maxi- a ( 
mum value of the abscissa for the oval — ; 
OABO is V27. It corresponds to a 7 vee: oe 
point A, where the tangent is parallel to . ee : 


OY, since the co-ordinates of this point 

reduce j",(x, y) to zero. Regarding y as 
an arbitrary variable, one finds that the 
maximum value of y for the same oval is V4. This maximum value gives 
the point B, where the tangent is parallel to the x-axis. 

The preceding method of discussion is applicable in all cases where the 
equation does not contain more than three terms; because it is always 
possible to determine the number of real roots of a trinomial equation 
involving one unknown quantity. . 








Fig. 215. 


THE INTRODUCTION OF AN AUXILIARY VARIABLE. 


341. When it is impossible to solve an equation with respect 
to one of the variables a or y, it is possible, in certain cases, to 
express the two co-ordinates in terms of an auxiliary variable ¢, 
and, on following the simultaneous variation of x and y, as ¢ 
varies between the limits which make these quantities real, to 
trace the curve. 

If y be regarded as a function of x, and # as a function of £, 
it follows that, on taking the derivative of y with respect to ¢, 


owing to the theorem of functions,* 
Dy = Dy» Dx; 
; Da 
whence it follows Diy => J : 
Dx 
* To designate the derivative of a function, the letter D is frequently 
employed, representing a partial derivative, and the variable with respect 





438 PLANE GEOMETRY. BOOK IV. 


which gives the angular coefficient of the tangent at the point 
which corresponds to any value of ¢. The values of ¢ which 
reduce D,y to zero determine the points at which the tangent is 
parallel to the x-axis, and the values which reduce D,a to zero, 
the points at which the tangent is parallel to the y-axis. 


342. ExampeLe V.— Construct the curve yt — y3x + 23 — Qa2y = 0. 
If we put y = ¢a, it follows that | 


2t—1 2t¢—1 
Lo ae (0p see Ee 
t3(¢ — 1)’ ae (t — 1) 


The curve is constructed by allowing ¢ to vary from —o to +o. In 
order to follow the variations of x and y, construct the derivatives 


_6P-8t+38 p,__4°-5t+2 


Dt = 
om t4(t — 1)? t3(t — 1)2 








The numerators do not become zero for any real value of t, and do not 
therefore change in sign. ‘The values of « and y become zero for ¢ = }, 
infinity for ¢=0 ort=1. If ¢ vary from —o to 0, a is negative and 
decreases from 0 to — », y is positive and increases from 0 to 0; thus 
> the infinite branch OA is obtained 

v. - (Fig. 216). As the variable ¢ increases 
from 0 to 3, x and y are positive and 
decrease from o to 0, which gives the 
* infinite branch BO. As the variable t 
increases from 34 to 1, x and y are 

“ negative and decrease from 0 to — o, 
L which gives the infinite branch OC. 
The angular coefficient of the tangent 
to the branch BOC at O is}. Finally, 
if ¢ vary from 1 to », # and y becom- 
C ing positive and decreasing from o to 
haat 0, one obtains the infinite branch DO. 

If the equation in # and y do not involve more than two groups of 
terms, one of the degree m, the other of the degree m — 1, and if the ratio 


I: 





a) 
be 





Y — + be chosen as an auxiliary variable, the co-ordinates x and y are 
. 


rational functions of this variable. If the equation contain three groups 





to which the derivative is taken is indicated by writing this variable as an 
index to the right and a little below the letter D. Thus, Dx and Dy 
indicate the derivatives of the functions « and y with respect to the 
variable t, D,y the derivative of y with respect to 7. 


CHAP. I. CURVES IN RECTILINEAR CO-ORDINATES. 489 


of terms, the first of the degree m, the second of the degree m — 1, the 
third of the degree m— 2, on using the same auxiliary variable, the 
co-ordinates may be expressed by the solution of a quadratic equation, 
and their simultaneous variations can still be followed. 


ExampLe VI. — Construct the curve x y* — cy -x—2=0. 
ig 
a 2 y= =! xamine how 
t#— 1 t+2 
x and y vary when the auxiliary variable ¢ varies from — 0 to + o. For 


this purpose construct the derivatives of the two functions ; one has 





If one put xy = ¢, it follows that «= 


_8t4+8641 
Mare eo me 


eS ks oe alent 


D,% = 
(¢ +2)? 


» Dy 








The value of D,xz becomes zero for two values a and 6 of f, comprised, the 
first. between ae and — 2, the second between —1 and 0. The value 


vo 
Dy becomes zero for three values c¢, d, e, of t, comprised, the first between 
-) and — 2, the second between — 2 and 0, and the third between 0 and 


1. It follows from the preceding that a<c,d<b. 
Now let us consider the following series of quantities 


—o,a,c, —2, —1, d,b, e, +1,0, 


arranged in order of magnitude. If ¢ vary from —» toa, & is negative, 
begins with 0, and decreases; y is positive, begins with infinity, de- 
creases continually; one ob- 
tains the branch AB, asymptotic 
to the y-axis (Fig. 217). The 
variable ¢ varying from @ to c, 
gz is negative and increases, y 
is positive and decreases ; one 
obtains the branch BC. The 
variable ¢ varying from c¢ to 
— 2, x is negative and increases ety 
to 0, y is positive and increases 
to o; whence the branch CZ. 
The variable ¢ varying from — 2 
to —1, z increases from 0 to o, 
y increases from —o to 0; 
whence the doubly infinite 
branch DE, asymptotic to OY’ and OX. The variable ¢ varying from 
—1 to d, x begins with — and increases, y begins with 0 and in- 
creases; whence the infinite branch FG, asymptotic to OX’. As the 
variable t varies from d to b, x continues to increase, and y remaining 
positive decreases; whence the branch GH. As the variable ¢ varies 














D 
Fig. 217. 


440 PLANE GEOMETRY. BOOK IV. 


from } to e, x and y decrease; whence the branch HK, which intersects 
OX’ in a point, whose abscissa — 2 corresponds tot = 0. As the variable 
t varies from e to +1, x decreases, y increases; whence the infinite 
branch AL, asymptotic to OX’. Finally, as ¢ varies from + 1 to + 0, 
x decreases from to 0, and y increases from 0 to ; whence the double 
infinite branch MN, asymptotic to OX and OY. 

The tangents at the points C, G, K, which correspond to the values Cc; 
d, e, of t, which reduce D,y to zero, are parallel to OX ; the tangents at 
the points B and H are parallel to the y-axis. 


343. Tancent Curves, OrtTHoGonaL Curves. — Let 
J (x, y) =9, (a, y) =0 be the equation of two curves. Call 
«x and y the co-ordinates of a point of intersection of the two 
curves; in order that they be tangent at this point, it is neces- 
sary and sufficient that the angular coefficients of the tangent 
to the two curves at this point be equal: 


anes 
i 
or SP 'y —f'y?'s 7 4) 


The co-ordinates « and y ought therefore to satisfy the three 
equations : 


(1) I (2, y) = 0, (a, y) =9, To, —fyo'2=9; 


on eliminating w and y between these three equations, that is, 
on expressing the condition that they have a common solution, 
we get an equation which expresses the condition that the 
curves are tangent at the same point. 

If the curves should be tangent in & points, it would be 
necessary to express the condition that the equations above 
have k common solutions. 

From a geometric point of view, to express the condition 
that equations (1) have k common solutions is equivalent to 
expressing the condition that the curve 


pak ae —f'yb'. = O 
passes through k of the points of intersection of the given 
curves. 


CHAP.I. CURVES IN RECTILINEAR CO-ORDINATES. 441 


In a similar manner it follows that if the given curves f= 0, 
¢ = 0 be orthogonal at a point (#, y), one ought to have 
(2) I (a, y= 0, ¢ (a, y= 0, S' P's +f',¢', = 0. 
On eliminating # and y between these relations, we get a con- 
dition that the two curves are orthogonal at one of their points 
of intersection. In order to express the condition that they 
are orthogonal at k of their points of intersection, it 1s neces- 
sary to express the conditions that equations (2) have & com- 
mon solutions. 

From the geometric point of view this amounts to expressing 
the condition that the curve 


St 9's + f'b', one! 0 


passes through & of the points common to the proposed curves. 


ExAMPLe. — Suppose that we have two conics f=0, ¢=0; in order to 
express the condition that they are orthogonal at their four points of in- 
tersection, it is necessary to express the condition that the curve 

I'x¢'e + f'yb'y = 0, 
which is also a conic, passes through the four points common to the two 
given conics ; that is, that its equation can be identified with an equation 
of the form f+ Ap = 0. 

Thus it may be easily verified that, whatever be the constants a and 
B, the two conics 

222+ y2—-ax=0, y2?—2px =0, 
are orthogonal at all of their points of intersection. 

The same is true of the conics 


2Qey—a=0, 2-—y?—p=0. 


Exercises. —1° Construct the general equation of the conics which 
intersect at right angles the fixed conic Ax? + By? —1=0 in four points. 

These conics may be divided into several groups; to one of them 
belong the conics confocal to the fixed conic. 

2° Let f(x, y)=9, o(@, y) =0 be the equations of two algebraic 
curves of degrees m and n in rectangular co-ordinates ; Jind the angle at 
which the curves intersect. 

Let x and y be the co-ordinates of one of the points of intersection ; 
the tangents to the two curves at this point intersect at an angle @, whose 
trigonometric tangent is given by the formula 
S'ch'y —f'yb's 
Soh atSyP'y 





tan:¢@ = 


4492 PLANE GEOMETRY. BOOK IV. 


By eliminating x and y between this equation and the equation of the 
two curves, one gets an equation of the degree mn, giving the tangents of 
the angle 6, at which the two curves intersect. 

As an application, form the equation of the second degree which 
determines the tangents ¢ of the angles at which the straight line 


yl . 2 
ux + vy + w = 0 intersects the ellipse - + 5 —-1=0. 


One finds : 


t2(a2v2w? + b2u2w? — ctu2v?) + (a2u? + b2v? — w?) (2 te2uv — a?u? — b?v?) =0. 


CHAP. Il. CONVEXITY AND CONCAVITY. 443 


CHAPERE 11 
CONVEXITY AND CONCAVITY. 


344. Let AB be an arc of the curve corresponding to a 
determination of y and to the values of # comprised between 
a and b; we assume that the second derivative y" of y with 
respect to # preserves the same sign in this interval, for 
example, remains positive. Draw the tangent RS at any 
point M of this arc, whose abscissa is 2; let y') be the value 
of the derivative at this point, or the angular coefficient of the 
tangent, and designate by Y the ordinate of any point of this 
straight line, the ordinate defined by the equation 


Y— Yo = Y'y (@ — Xp) 5 


the difference y — Y becomes zero for «= a, and the same is 
true of its derivative y!— Y' or y'—y) (Fig. 218). When the 
abscissa 2 increases from a to 0, the 
derivative y' of the difference y'— y‘ 
being positive, the function y'—y') in- 
creases; since it becomes zero for # = %, 
it is negative from @ to a, positive from 
a, to b. Consider now the function 
y — Y, which has the derivative y! — y'o; 
when 2 varies from a to 2, the deriva- 
tive being negative, the function de- 
creases; since it becomes zero for # = a, it would be positive 
above; as x varies from a to b, the derivative is positive and 
the function increases; since it becomes zero for %== % it 18 
also positive from a to 6, whence it follows that the differ- 
ence y — Y remains positive throughout the interval from a to 
b. We conclude from this that the are of the curve ab is situ- 
ated wholly on the same side of each of its tangents, and is 
said to be convex. Similarly, if the second-derivative were 





Fig. 218. 


444 PLANE GEOMETRY. BOOK Iv. 


negative; the difference 7 — Y being negative, the are would 
be situated wholly on the other side of the tangent. It is easy 
to distinguish these two cases; draw through the point Ma 
straight line WY, parallel to the axis OY, and in the direction 
of the positive y’s; in the first case, the arc is situated on the 
same side of the tangent as. the half-line MY,; in the second 
case, the are lies on the other side. In the first case, it is said 
that the are AB is concave in the direction of MY; in the 
second case, in the opposite direction. 

We know that the sign of y" indicates the kind of variation 
of y' when 2 increases. If therefore one imagines that. the 
point M travels through the are AB, the angular coefficient 
will increase if y'" be positive, and, on the contrary, will 
decrease if y"’ be negative. 


345. The points of a curve at which its concavity changes 
its direction are called points of inflection. At such points 
therefore the second derivative changes its sign. The second 
derivative may change its sign on passing through zero or 
infinity. In general, the quantity y'', being finite and continu- 
ous, changes sign on passing through zero. Suppose that y"! 
changes its sign on passing through zero 
for «= 2%, it may be verified that the first 
derivative y'—y') does not change sign, 
but that the function y—Y does. experi- 
ence a change in sign; of the sort that at 
this point the curve passes from one side 
to the other of the tangent. If for v= ay, 
y'' experiences a change in sign on passing 
through infinity, y’ becoming infinite and y 
remaining finite, the point «=a and 
Y= Yo is a point of inflection at which the tangent is parallel 
to the y-axis. 

If a neighboring secant of the tangent WT be drawn through 
the point of inflection M, this secant will intersect the curve 
in two points M' and M"; the tangent M7’ is the limit of 
the secant passing through the three points M", M, M' when 
the two points M" and M' approach the point M. 





Fig. 219. 


CHAP. II. CONVEXITY AND CONCAVITY. 445 


346. Exampre I. — Sinusoid. Construct the curve y=sing. <As 
x increases from 0 to z, the ordinate is positive; it begins with 0, 
increases to 1, and then decreases to 0, which gives the are OAC 


(Fig. 220) symmetrical to the ordinate which corresponds to x = - AS 


x increases from 7 to 27, y becomes negative, and one obtains the arc 
CBO’, equal to the first. From 27 to 47, the ordinate passes again 
through the same values which it took when z varied from 0 to 27, 


Y} Lb 5 


A A 


a ir a eee ee 
| G Be ae SS 


\ 




















Fig. 220. 


similarly from 4 to 67, etc. Thus the curve is composed of an in- 
finite number of equal undulations. 

The angular coefficient of the tangent is y’ = cosx; at the origin, 
y' =+1, and the tangent is the bisector of the angle YOX. At the point 


C, y' =—1, and the tangent is parallel to the other bisector. For «= = 


od 


the derivative becomes zero and changes from positive to negative; the 
ordinate of the point A is a maximum. For “= 35 the derivative 


‘becomes zero again and changes in sign from negative to positive ; the 
ordinate of the point B is a minimum. 

When x varies from 0 to 7, the second derivative y!! =—sinz is 
negative, and the concavity of the curve is turned toward negative y’s ; 
from r to 27, the second derivative is positive, and the concavity of the 
curve is reversed and turned toward positive y’s ; the point C is therefore 
a point of inflection. : 

It is worthy of notice that the curve has an infinity of centers, situated 
at equal distances along the «-axis, the points O, C, O', --- whose abscissas 
are multiples of . Each of them is a point of inflection. 


347. Examere II. — Construct the curve (y — x2)2 9 =, or 
5 
y = 2 +4 er 
The values of y are only real when 2 is positive. Consider first the 
5 
case when the sign before the radical is + ; the functions y = «? + 2? 


446 PLANE GEOMETRY. BOOK IV. 


increases Oi 0 to o, as x varies from 0 to o. The derivative 


f= 22543 5 oo begins with zero and increases continually without limit ; 
one gets therefore one infinite branch OD a 

* gent to the x-axis at the point O, and which 
has its concavity turned toward positive y’s 

(Fig. 221). Investigate now the minus sign 

before the radical; the value of y is positive 

from 0 to 1, and negative for «>1. Lay off 

AN on OX a length OA equal to unity ; the curve 
“ah pases through A. The derivative y’ =2a 


‘¢ 


-—, 





M 
\ 
{ 
Pp 





Fig. 221.’ 
5 at is zero at the point O, remains positive 


so long as xis less than 38, oa becomes negative when « is greater than 
this number ; the ordinate MP, which corresponds to }%, is a maximum, 
and the eugene at M is parallel to the z-axis. The second derivative 


2— 15 42 remains positive for 02 2< 4%;4, but is negative for «> iS ; 
the point N, which corresponds to the alee #s';, is a point of inflec- 
tion; from O to N, the concavity is turned toward the positive y’s, but 
for «> $4, it is turned toward the negative y’s. 

The two branches of the curve are tangent to the x-axis at the point 
O, without one being the continuation of the other; points which have 
this peculiarity are called cusps. In this curve, the two branches lie 
on the same side of the tangent. On considering the curve (y — 2)? 
— #3 = 0, there will be two branches, one situated on one side and the 
other on the opposite side of the tangent ; the cissoid has a cusp of this 
kind at the vertex (Fig. 16). 


348. Examp te III.— Let the curve be y =5 (ea +e 2), 


It is supposed that @ represents a given length ; then the equation is 
homogeneous and defines a curve, which is called the catenary, for the 
reason that it is the curve of the form assumed by a flexible thread of 
which the ends are attached to two fixed points. 

The equation gives equal values of y for equal values of x with contrary 
signs; hence the ee line OY is an axis of ye curve. If # vary from 


0 to w, the term e« increases, but the term e a decreases ; in order to 
know how y varies, construct the derivative 

c" i ¢ of y with respect to x; namely, 

= sad 

y! = 4(ee —@ 2). 
This derivative is positive for all positive 
values of x; therefore when x increases from 
0 to «, the value of y increases constantly 
from ato o, which gives an infinite branch 
Fig. 222. BC (Fig: 222); the branch BC’, the sy- 








CHAP. I. CONVEXITY AND CONCAVITY. 447 


métrique of BC with respect to OY, is obtained by assigning negative 
values to x. 

Since the second derivative remains positive as # varies from — o to 
+, the curve turns its concavity toward positive y’s. 


1 
349. Examete IV.— Construct the curve y= ez. For very small 


positive values of 2, y is positive and very large ; as x increases from 
0 to +0,y decreases constantly from to 1, which gives a branch 
AC (Fig. 228), asymptotic on the one hand to the y-axis and on the 
other to the straight line G’G drawn parallel to the w-axis and at a 
distance from this straight line equal to 











unity. When one gives a very small zie 
numerical negative value to x, y is posi- 
tive aud very small; as 2 varies from 0 
to —o, y increases continuously from 
0 to 1, whence we get a branch OD . CG 
starting from the origin and asymptotic D ne G 
to the straight line GG’. N\ 0 

This curve presents a peculiarity id Ne at 
which has not yet been met: the 

Fig. 228. 


branch DO stops abruptly at the point 
O; points of this kind are called points @arret. 
In order to find the direction of concavity, construct first the first 


1 
derivative y! = — Je. When x varies from 0 to 0, the two factors = 
1 x x: 


and ez diminish, and on account of the sign —, y’ increases; the con- 
cavity of the branch AC is turned toward positive y’s; since, as x varies 
1 


from —o to 0, the factor a increases, and the factor e« diminishes, it is 
x 


not at first evident how y’ varies and we construct the second derivative. 
1 


We get y= As « varies from —o to —}, the second deriv- 


at 
ative is negative; take OP = } and let M be the corresponding point of 
the curve ; the arc DM is concave toward the negative y’s; as x increases 
from — 3} to 0, y’’ is positive and the are M O turns its concavity toward 


the positive y’s and the point M is a point of inflection. 


Wen 
Exampte V.—Let y=evY*. 
ae 
a- Va — 2 
Pdi Aide 
5 

9x5 

the second derivative changes its sign twice, once on becoming infinite for 


. ° 3 
z= 0, and a second time on becoming zero for Vx =2, x=8. The 
curve has therefore two points of inflection. 


It can be easily shown that y!’ =e 


448 _ PLANE GEOMETRY. BOOK IY. 


350. Consider the cases when the equation 


(1) S(@, y) =9 

cannot be solved with respect to either of the variables x and 
y; the derivative y' is given by the equation 

(2) SA, YAS @ Y) + y'=9. 


Since y and y’ are both functions of a, the first member of 
equation (2) is a compound function with respect to the inde- | 
pendent variable x; the derivative of this function is 


Se + ZH yy +S yy? + uP eae 


since it is always zero, its derivative is also zero, and we have 
the equation 


(3) Satay! + fi ypyPe + fly" = 0, 


which determines the value of y'. If in this equation y' be 
replaced by its value deduced from equation (2), (3) becomes 


fi Mie aa Mee 
Cis 


It is by means of this formula that the direction of concavity 
and also the points of inflection are determined. 





351. Apply this formula to the curve defined in § 339. The equation 


of this curve can be written in the form 
t 


(1) Fr Y= ELG? + y? +o)? — 4 a? — at] =0. 
From (1) we deduce 
Se=eeP+yP—e), fy=yeV+y+e), 
Sa = (27 + y* — a 2 x, Izy = 22Y, Jy = (24 + y? + c7) +2 y?. 
If these values be substituted in the preceding formula, we obtain, after 
reduction and using formula (1), 
__ at[8 c2(y? — #2) — (at — e*)] 





yl! 


Since for each portion of the curve comprised within one of the angles of 
the co-ordinate axes, the denominator preserves the same sign, the value 
of y'’ cannot change sign other than when the numerator passes through 


CHAP. II. CONVEXITY AND CONCAVITY. 449 


zero. The co-ordinates of the points of inflection should therefore satisfy 
at the same time equation (1) and the equation 








i. 9 oan 
(2) y—2—- TF =9; 
whence it follows that 
at _ae ct 
(3) y? + x a 3 ° 


In the first case, when a is less than c, the values of « and y given by 
equations (2) and (3) being imaginary, the numerator of y'’ has the same 
sign for all points of the are BCA; it may be easily verified that this 
numerator is negative for the points B and A; the concavity of this arc is 
directed toward the negative y’s (Fig. 211). In the second case, a=c, the 
numerator of y!! becomes zero at the point O only ; it is negative from O 
to A, and the concavity of the are OCA is directed toward the negative 
y’s (Fig. 212) ; the are A!D!OCA has a point of inflection at the point O. 
In the third case, one has a>c; here the values of « and of y are real 
and one has at the same time a<cv2. If a be greater than cv2, the 
numerator is negative for all the points of the arc BA, and the concavity 
is directed toward the negative y’s (Fig. 214) ; if a be less than cV2, the 
numerator becomes zero for a certain point G (Fig. 213) situated between 
Band C; from B to G, it has the same sign as at the point B; it is posi- 
tive and the concavity is directed toward the positive y’s; from G to A, 
the numerator has the same sign as at the point A; it is negative and the 
concavity is turned toward the side of the negative y’s. ‘The point Gis a 
point of inflection. 


REMARKS CONCERNING ALGEBRAIC CURVES. 


352. Let f(a, 7)= 0 be an integral, algebraic equation of the 
degree m with respect to w and y, and of the degree n with 
respect to y; to each value of x there correspond n values of 
y, which, in general, are different from one another; it can be 
demonstrated that, when # varies in a continuous manner, each 
of these values varies also in a continuous manner; we assume 
this theorem as we have done in previous discussions. When 
the equation is irreducible, it cannot have multiple roots except- 
ing for a limited number of values of 2; among these values of 
a, consider only those which are real and suppose them arranged 
in order of magnitude. Let a,b,c be three consecutive values; 
when 2 varies from a to b, the number of real values of y will 
remain the same; because if, in the interval, an imaginary root 

2F 


450 PLANE GEOMETRY. - BOOK IV. 


should become real, its conjugate would also become real, and, 
at the moment of transition, the two roots would become equal ; 
one obtains also, in the interval considered, a certain number 
of real and distinct branches which do not have a common 
point. When 2 passes through the value }, it can happen that 
two real roots become imaginary or conversely; at the point 
which corresponds to the value } and to the real double root, 
two branches of the curves begin or end in this case. 

Among the real values of y which correspond to the same 
value a, of x, consider a value y which is a simple root; if x be 
allowed to vary from a — h to x + h, h being sufficiently small, 
this value of y will remain real without becoming equal to any 
of the others and will give rise to a real branch. Thus, when, 
for a value of x) assigned to x, the equation has a simple real root 
Yo, there passes through the point whose co-ordinates are x) and Yo 
one real branch, and only one. 

Let us consider next a value «= 0, to which there corre- 
sponds a multiple value y, of the order p. Locate the point 
M, whose co-ordinates are x = b, y= 4y,; among the p values of 
y which become equal to y, for «= 6, there are a certain num- 
ber which were real and a certain number which were imagi- 
nary; the number of the latter being even, the number of the 
real roots is p—2q (q can be zero). Similarly, when & varies 
from b to ¢, the number of real values of y which belongs to 
the value y, for v=b is p —2q'; of the sort that the total 
number of branches of the curve which emanate‘ from the 
point M, in one direction or another, is the even number 
2p—2q—-2¢'. 


353. Let us determine the tangents at the point M; trans- 
fer the origin of co-ordinates to this point, and put y= ta; 
we will have an equation ¢ (a, t) = 0, which will determine the 
angular coefficients ¢t of the secants drawn from the point M 
to the points where the curve is intersected by a parallel to 
the y-axis. Suppose that for «=0 one has a real root t= 4; 
this root will determine a straight line, and, on repeating the 
same reasoning of the preceding paragraph, one sees that 
the total number of real branches emanating from the point 


CHAP. II. CONVEXITY AND CONCAVITY. 451 


M and tangent to the two directions of the straight line, is 
even. 

It follows from what precedes that an algebraic curve can- 
not have a point d’arrét (§ 349). It cannot have, moreover, a 
point saillant or anguleux; a point anguleux is a point at which 
two branches are tangent to two different straight lines. 


354. When the origin is transferred to the point M, whose 
co-ordinates are v% and y, the equation becomes 


(1) (af, oH uf, + 1 (a? re + 2 ay ne te Pf" ya) + + == 0, 
and the equation 
2 
(2) af, +, ety, +2¢f", ef s)+ --=0 
was uae _ 0 0% 0 


gives the points in which any straight line y = ta drawn from 
the point M intersects the curve. When one of the first 
derivatives at least is different from zero, the root 7 =0 being 
a simple root, the point Wis called a simple point of the curve. 
For the particular value ¢, of ¢ which reduces the first term to 
zero, a second root is equal to zero, and the straight line 
becomes a tangent to the branch of the curve. The contact is 
of the first order if for ¢=¢, the coefficient of 2 be different 
from zero; it is of the order p if the first coefficient different 
from zero is that of w?*'; among the m — 1 other points of in- 
tersection of the straight line and of the curve, p coincides 
with the point ©. 

Suppose that the two partial derivatives of the first order be 
zero, without the three derivatives of the second order being 
zero; the root «= 0 being a double root of equation (2), any 
straight line drawn through the point M intersects the curve | 
in two points coincident with M, and this point is called a 
double point. If all the terms be divided by 2”, equation (2) 

reduces to 


8) EM gt 2M ag HOM) +5 gM ag t 8 Tay t i) be 0. 
When the equation of the second degree 
(4) Fi + 2if' he + ef" = De 


452 PLANE GEOMETRY. _ BOOK IV. 


has two roots ¢t, and ¢, real and unequal, for a very small abso- 
lute value of x, equation (3) has two consecutive simple roots, 
t; and t,; to these real values of t correspond two branches of 
the curve tangent to the straight lines y=t, y=t. (Fig. 
208). If equation (4) have two imaginary roots, the two values 
of t which are consecutive are also imaginary, and the point W 
is an isolated point (Fig. 207). When equation (4) has its two 
roots equal to ¢,, several cases can arise; if the two values of ¢ 
consecutive to ¢; are imaginary for the positive and negative 
values of 2, the point M is an isolated point; if they are real 
for the positive values of x, and imaginary for negative values 
or conversely, one has a cusp (Figs. 209 and 221); finally, if 
they are real for positive values, and also for negative values 
of a, one has two branches passing through the point M, from 
one side to the other, and tangent to the same straight line. 

We notice that the equation which gives the various tangents at 
a multiple point is obtained by equating to zero the group of terms 
of lowest degree in equation (1). 


354. 2. We proceed to define a case in which it is easy to 
find the form of an algebraic curve in the neighborhood of one 
of its points. . 

This point being taken as origin, one supposes that the equa- 
tion in 2, found by making y = 0 in the equation of the curve, 
has zero for a simple root. Let 


$o (2) + dila)y + delay te + $,(ay" +> =0 


be the equation of the curve written in an integral form and 
arranged with respect to increasing powers of y. By hypoth- 
esis $) contains x as a simple common factor; x can also be a 
factor of some of the coefficients following qd). Let $, (a) be 
the first coefficient which does not become zero for x = 0, then 
the equation can be written 


w@ fo (@) + yr (a) +e Fy" "Wn a(@)} 
+ y” Sb, (@) + Ybn41(@) + aah } — 0. 


If x be supposed very small, and if one of the very small 
values of y be considered, the sign of each of the parentheses 


CHAP. II. POINTS OF INFLECTION. 453 


is the same as the sign of its first term, and each of these 
terms, ¥%(«), $,(x), can be replaced by the value which it takes 
for ss; 

Finally, in order to find the form of the curve in the vicin- 
ity of the origin, the equation being arranged with respect to 
increasing powers of y, it is sufficient to consider the term in- 
dependent of y, which contains # as a simple factor, and the 
first of the terms following it whose coefficient does not 
become zero for x= 0. The question is reduced to the consid- 
eration of a binomial equation 


Ax + By" =0 


and one can similarly, in each of the coefficients A and B, 
neglect the part which becomes zero for «= 0. 


355. One can therefore, in seeking the equation of the tan- 
gent to an algebraic curve and the points of inflection of this 
curve, employ the following method, which has the advantage 
of applicability to the curves whose equation is given in tri- 
linear co-ordinates. 

Consider a curve of the order m whose equation in homo- 
geneous co-ordinates is f(a, y, z)=0. Take on the curve a 
point M,(a,, %, 2;) and in the plane a second point M(a, y, 2). 
Seek the points where the straight line 14M, which joins these 
two points, intersects the curve. The homogeneous co-ordi- 
nates of a point of this straight line are (§ 330) 


(5) M+AX, YWtrAY, m+AZ; 
in order that this point belong to the curve, it is necessary and 
sufficient that A satisfy the equation 
Stra, ytdry, mt+Az) =, 
or (6) f(y Yay 1) FACP, + Un + Pn) 
r? 

1-2 

On substituting successively all of the roots of this equation 


in \ into the expressions (5), one will obtain the co-ordinates 
of all of the points of intersection. The point (q, Yy, %) 





7 (a "ot yf ate) te =O. 


454 PLANE GEOMETRY. BOOK Iv. 


being on the curve, one has f(a, %, 21) =0; equation (6) has 
therefore as a root A = 0, which, substituted in expressions (5), 
gives no other than the point (a, y, 2). In order that the 
straight line M,M be tangent to the curve in M, it is necessary 
that it have two points of intersection coincident with ™M,, 
that is, that equation (6) have the double root A? = 0: the con- 
dition for which is 


(7) fie + Uy, + fs, = 0. 


If the co-ordinates of the point M satisfy this equation, the 
straight line 4M is tangent at M,: equation (7), in which one 
considers x, y, z as the current co-ordinates, is therefore the 
equation of the tangent at the point M,. 

In what precedes it has been assumed that the three deriva- 
tives f',, Jy) fz, are not zero at the same time. If these three 
derivatives were zero, the coefficient in \ in equation (6) would 
be zero, whatever be the position of the point M: every straight 
line passing through the point M, would intersect the curve in 
two points at least coincident with M,. It is then said that the 
point M, is a singular point. The singular points are therefore 
characterized by the following property, that their co-ordinates 
reduce the three partial derivatives of f with respect to a, y, 
and z to zero, and, consequently, f, by virtue of the theorem 
of homogeneous functions. 

Suppose that the point M, is not a singular point, and seek 
the condition in order that it be a point of inflection. For 
this purpose it is necessary and sufficient that the tangent at 
M, intersect the curve not in two but in three points coincident 
with M,; in other words, if the point M be taken on the tan- 
gent (7), it is necessary that the coefficient A” become zero, that . 
is, that one has 


(8) P(X, Y, z= HF", a YS ye ToT 2 i. 2 yf ye, 

+ 2 20f", 2 +2 wnuf", : 
If x, y, 2 be regarded as fhe current co-ordinates, equation (8) 
represents a conic; and since every system of values 2, y, 2 


satisfying condition (7) ought to satisfy at the same time condi- 
tion (8), this conic should resolve itself into two straight lines 


2, 


CHAP. II. POINTS OF INFLECTION. 455 


one of which is a tangent. Therefore if M, be a point of in- 
flection, the discriminant of the function of the second degree 
d (a, y, 2) ought to be zero. 
Bae 2 7 Bd 
cua f',2 Bue oe 


! tf ‘ki 
2 
og od Y474 i =t 


Conversely, if at a non-singular point M, this function H be 
zero, this point is a point of inflection. In fact, we notice that 
the conic $ (a, y, 2)=0 passes through the point (a, y,, 2) and 
is tangent to the curve. It passes through the point, because, 
owing to the theorem of homogeneous functions, 


(M4, ry a)= co te fee min — 1S Wy a)=9; 


then, owing to the expression for ¢, it may easily be shown 
that 
vi’, + Wo', Zp, ais 2 (m oat 1)(af",, a3 uns. ae t's) 


identically, which shows that the tangent to the conic at the 
point M, coincides with the tangent to the curve at this point. 
Therefore, if the conic ¢ decompose into two straight lines, 
one of them ought to pass through M, and be tangent to the 
curve; in other words, the polynomial ¢ (a, y, 2) is resolvable 
into two factors of which one is the first member of (7), of the 
equation of the tangent. It follows that the point M, is indeed 
a point of inflection. 

The determinant H becomes also zero when the point M, is 
a singular point; in fact, in this case the three partial deriva- 
tives of the polynomial d(x, Y, 2)> b'» by o'. become ose for 
B= My Y= Yay ¥= By since it follows from this that f',, S's, 
become zero at a singular point; the discriminant H eu % is 
therefore zero. 
- The determinant H is called the Hessian; if %, %, % be 
regarded as current co-ordinates, equation H7=0 represents a 
curve of the order 3(m—2) called Hessian which passes 
through the points of inflection and the singular points of the 
curve f=0. We shall see that, conversely, every point com- 
mon to the two curves f=0, H=0, which is not a singular 


* 


456 PLANE GEOMETRY. BOOK Iv. 


point of f= 0, is a point of inflection. Whence one concludes 
that, if the curve f=0 do not have singular points, it has 
3m(m—2) points of inflection real or imaginary. If the 
curve f= 0 have singular points, the number of its points of 
‘inflection is diminished. | 

Thus a curve of the third degree without a singular point 
has nine points of inflection; if it have a double point, it has 
no more than three points of inflection; if it have a cusp, it has 
no more than one. 


Exampie. — The curve of the third order which has in Cartesian co- 


ordinates the equation 
A*+Y*+1=0, 


or in homogeneous co-ordinates 
S (@y y, 2) = #8 + y® + 29 =0, 
does not have singular points ; because the three partial derivatives 
fr=8e2, fy =S8y%, fl, = 822 


do not become zero for any system of values of x, y, z which are not all 
three zero. Here the Hessian is 


if = OF ag = 0, 


or xyz = 0; it decomposes therefore into three straight lines x =0, y =0, 
z = 0, which are the two co-ordinate axes and the line at infinity. The 
nine points of intersection of the Hessian with the curve will be the 
points of inflection. One has, for the co-ordinates of these nine points, 
on calling w acubic imaginary root of unity, 


ae p 
x=0 with TES or oS or oS ws 
z Zz z 
y=0 with -=—1, or —-=~—a», or — = gy, 
z z z 
‘ y 
2=0 with fo or LES or —~=—¢’%, 
x x x 


The last three points are at infinity. The three points 


ae Af), v=-1; v= 0, =—1; a z=-1 


only are real. They lie on the straight line «+ y+ 2=0. 


Cuass or A Curve. — The class of a curve f(a, y, z2)=0 is 
the number of tangents which can be drawn from a point of 
the plane to this curve. Let M,(a,, y2, 2) be a given point; 


CHAP. IT. THE CLASS OF A CURVE. 457 


on expressing that the tangent at the point M,(a, y, %) pass 
through M2, one has the condition 

(9) Xf + + Yof ‘. + af fs = 0, 

which, combined with f(a, y, z,)=0, determines the points of 
contact of the tangents emanating from the point MM. Every 
non-singular point 4, whose co-ordinates satisfy these equa- 
tions is a point such that the tangent at this point passes 
through M,; equation (9) is, moreover, satisfied by the co-ordi- 
nates of all of the singular points, since these co-ordinates 
reduce f',, f',, f',, to zero. If x, y, % be regarded as current 
co-ordinates, equation (9) represents a curve of the order 
(m —1), called the first polar of the point M, with respect to the 
curve. 

Every point common to this curve and to the given curve 
f=0 is a singular point or a point of contact of a tangent 
passing through the point MM, These two curves have 
m(m—1) common points; if the proposed curve does not 
have singular points, these common points are all of the points 
of contact of tangents drawn from M;. Therefore, a curve of 
the order m without singular points is of the class m(m — 1). 

If the curve have singular points, the number of tangents 
emanating from M, is equal to m(m—1) less the number of 
the points of intersection of the polar (9) and of the curve, 
which are coincident with the singular points. 

On supposing that the singular points of the curve consist of 
double points or cusps, and designating by d@ the number of 
double points, by 7 the number of points which are cusps, by ¢ 
the number of points of inflection, and by c the class of the 
curve, the following formulas, due to Pliicker, may be demon- 
strated : 

c=m(m—1)—2d—3r7, 
t=3m(m—2)—6d—8r. 


ExamMpLe.—A curve of the third order without a singular point is 
of the sixth class. 

Take a curve of the third degree with a double point, for example 
a curve whose equation in Cartesian co-ordinates is 


(10) ¥24 X2(X—a)=0. 


458 PLANE GEOMETRY. BOOK IV. 


The tangents at the origin are given by the equation 
Y2—axX?=0; 


if therefore a be different from zero, the origin is a double point at 
which there are two distinct tangents (real or imaginary, according as 
a is positive or negative); if @ be zero, the origin is a cusp, and the tan- 
gent at this point is the z-axis. The equation rendered homogeneous is 


S (4, Y, 2) = yP2 + (x —az)=0; 
whence fle =322—2axz, fly=2yz, firx=y? — ax 
The first polar of the point M’(x’, y', 2’) has the equation 
eabiee x! (8 02 — 2axz)+ 2 y'yz + 2'(y? — ax?) =0; 


this polar is a conic passing through the double point situated at the ori- 
gin « = 0, y = 0, and having a tangent at this point whose equation is 


(12) . axa! + yy! = 0. 


If a be different from zero, the origin is a double point with distinct tan- 
gents, and the tangent (12) varies according to the position of the point 
M'. The polar conic (11) intersects therefore the curve of the third order 
in six points, two of which are coincident with the double point. There 
are therefore but four of these points of intersection which do not coin- 
cide with the singular point, and but four tangents can be drawn from 
the point M’. The curve is therefore of the fourth class. 

If a be zero, the origin is a cusp; the polar conic (11) passes through 
this point, and the tangent to the conic at this point has the equation 
y = 0 asa tangent to the curve. The polar conic is therefore tangent to 
the curve at the cusp: it intersects it in three points coincident with this 
singular point, and in three other points only, which are the points of con- 
tact of the tangents drawn from the point M’. The curve is therefore of 
the third class. ; 

One could, as an exercise, form the tangential equation of curve (10), 
that is, the condition that the straight line 


uxrX+vY+w-—0 


be tangent to this curve; it can be verified that this equation is of the 
fourth degree in wu, v, w, and reduces to the third degree when a is equal 
to 0. 


356. Curves IN TRILINEAR Co-oRDINATES. — Let 


F(a, B, y) =9 
be the equation in trilinear co-ordinates of a curve of the 


CHAP. Ii. TRILINEAR CO-ORDINATES. 459 


degree m. Take, on this curve, a point M,, whose trilinear 
co-ordinates are 0, 8), y:, and, in the plane, a point M(a, B, y). 
Find the points where the straight line M,M, which joins 
these two points, intersects the curve. The trilinear co-ordi- 
nates of a point of this straight line are (§ 331) 


+ Aa, Bit yB, y+ Ay; 
in order that this point belong to the curve, it is necessary and 
sufficient that A satisfy the equation 
F (a, +a, Bit rB, 1 os AY) = 
or F(o, By yi) + (ak, + BE, + y7F,,) 


2 
ds 5 (OF 4+ BF", » + see )ene = 0. 





x 


From this equation, which in every respect is similar to equa- 
tion (6), one deduces results identical with those which one 
deduced from equation (6). We find thus that: 
1° If the three partial derivatives FY» F's, F'',, be not zero, 
the tangent at the point 4, has the equation 
al + BE sg, + y= 95 
2° If the three partial derivatives #",, F",, F',, be zero, the 
point M, is a singular point; 
3° In order that the point M, be an inflection, it is necessary 
that the Hessian 
Baa iF ‘oP, ri ‘on 
H= Ia, Pg. tay ie = 0; 


" ' tf 
171 Byyi ¥1? 


and, conversely, if at a non-singular point M, the Hessian be 

zero, this point is a point of inflection. 
Exampve. — Consider the equation of the third degree 
F(a, B, y)= a3 + 62+ 73+ 6kapy =0; 


it can be demonstrated that the equation of every curve of the third order 
without a singular point can be reduced to this form by a suitable choice 
of the triangle of reference. 


460 PLANE GEOMETRY. BOOK IV. 


Here we actually have 


a ky =kB 
HO ky B ka 
k8 ka + 4 





or, on developing, 
HAH = 6° [— k? (a3 + 63+ 73) + (14 2%) aBy]. 


The points of inflection will be the points of intersection, nine in number, 
of the given curve / = 0 with the Hessian H=0. The equations F = 0 
and H = 0 are homogeneous equations of the first degree with respect to 
the two expressions a? + 634+ 3, aBy: the determinant of the coefficients 
of these two expressions is 1+ 8?; if therefore 1+ 8? be not zero, 
from the equations F = 0, H = 0 may be deduced the two following : 


aS + 68+ y8=0, apy =0, 
which give, for the nine points of inflection, 
a=Owith B+ y7=0, orB+wy =), or-B + wy =0, 
B=0 with y +a =0, or y + wa =0, or y + wa = 0, 
y =0 with a + B=0, ora + w8 =), ora + w*8=0, 


where w designates an imaginary cubic root of unity. These nine points 
are the same whatever k may be. It is easily seen that the straight line 
which joins two of these points passes through a third. 

If 1+ 8k3=0, the equations F=0 and H=0 represent the same 
curve ; then the Hessian coincides with the given curve: all points of this 
curve are points of inflection, which can only happen if it be composed of 


three straight lines. In fact, the equation k3 =— 1} gives for k three 
values : 
i es e 
i ee 
For k =— 1, the curve F = 0 becomes 


F=03 + 634 y3—3aBy= (a+ P+7) (at oB+ wy) (a+ WB + wy) =0; 


it is therefore resolved into three straight lines ; similarly if one put 
2 
k = = k = al 


as is seen on substituting, in the identity above, for a, wa or wa. 


461 


CHAPTER III 
ASYMPTOTES. 


357. When a curve has an infinite branch MN (Fig. 224), it 
can happen that the distance MH of a point M of this curve 
from a straight line CD approaches zero, when the point Mis 
continuously removed toward 
infinity; in this case, the 
straight line CD is called an 
asymptote of the branch of 
the curve. 

Consider the difference MR 
between the ordinates of the 
curve and of the straight line, 
which correspond to the same / 
abscissa, and let B be the angle 
which the straight line CD Fig. 224. 


makes with the y-axis; one has MR = ae 
sin 








; if either of the 





quantities MH and MR approach zero, the other will also 
approach zero. An asymptote can therefore be defined as a 
straight line such that the difference between the ordinates of the 
curve and of the straight line approaches the 








D 
limit zero when x is indefinitely increased. y : 
However, this definition is not applicable 
if the angle B be zero; that is, when the . ¥ = 
asymptote is parallel to the y-axis. In this 
case, if the straight line MR (Fig. 225) be arse aay 
drawn parallel to the w-axis, the straight of | 


line MR approaches zero, when the ordinate Fig. 225. 
increases without limit. If abe the abscissa of any point of 
the straight line CD, the abscissa of a point M of the branch 


462 PLANE GEOMETRY. BOOK IV. 


MN approaches a when y is increased without limit, or con- 
versely, y increases without limit, when a approaches a. 


ASYMPTOTES WHICH ARE PARALLEL TO THE y-AXIS. 


358. According to the preceding, the asymptotes of this 
species are obtained by seeking the finite values of a2 which 
render one of the values of y infinite. When the equation of 
the curve is solved with respect to y, one perceives, generally, 
these values at once; as examples we cite the cissoid and the 
strophoid discussed in §§ 20 and 23. 

If the equation be algebraic, but not solvable with respect to 
y, we proceed in the following manner. Let m be the degree 
of the equation, n the largest exponent of y; the equation can 
be written in the form 


hy (x)y” + ¢$, (x)y*—" + do (xy +---=0, 


do $1, bo +++ representing polynomials in x, whose degrees are 
at most respectively equal to m—n, m—n+1,m—n+2,++, 
and after dividing by y to the nth power, 


ce 1 1 
f 0 1 (2) = + gy (") = + + , (2) —= 0. 
(1) bo(#) + 1 (#) 7 + $2(@) a + +¢ ®) 7 


Suppose that a real branch MN be asymptotic to a straight 
line CD parallel to the axis of y and having the equation 
a=a. As the point Mis removed to infinity on this branch, its 


abscissa x approaches the finite value a, while : approaches 
y 


zero. Since the terms of equation (1) beginning with the 
second approach zero, it follows that the abscissa a reduces the 
polynomial ¢)(a#) to zero. Whence the abscissas of asymptotes 
parallel to the y-axis satisfy the equation (x) = 9. 


359. Conversely, let a be a real root of the equation bo(a’) =. 
it is necessary to examine, if it have real branches which 
approach continually the straight line «=a and how many 
there are of such. Suppose in the first place that a be a single 
root of the equation (x) = 0. 


CMAP xIt: ASYMPTOTES. 463 


If w be regarded as a function of : given by equation (1), 
when - approaches zero, it being either positive or negative, 
y 


one value of x, and only one, approaches a; this value of x 
is necessarily real; because, if it were imaginary, the conjugate 
root would also approach the real quantity a, which would then 
be a double root. One infers therefore that there exist two 
real branches asymptotic to the straight line «=a, one on the 
side of the positive y’s, the other on the side of the negative 7/’s. 

Suppose now that a be a double root of the equation ¢)(7) =0. 


When : approaches zero, two values of x approach a; these 
2 
values can be real or conjugate imaginaries. If they be real 


for very small positive values of s there exist two real 


branches asymptotic to the straight line «=a on the side of 
the positive y’s. If they be also real for very small negative 


values of = there exist two other real branches asymptotic to 


the same straight line on the side of the negative y’s. When 
the two roots are imaginary for the positive or negative values 


of = there does not exist any real. branch asymptotic to the 


straight line «= a. 
In general, let p be the order of the root a; among the p 


values of 2, which approach a when approaches zero, p—24q 
are real for very small positive values of 7 p —2q' for nega- 


tive values. There will be p—2q real branches asymptotic 
to the straight line «=a on the side of the positive y’s, and 
p—2q' real branches asymptotic to the same straight line on 
the side of the negative y’s; in all, 2p —2q—24q' real branches 
asymptotic to the straight line «=a. It is worthy of notice 
that this number is even. 


359. 2. In the particular case where a is a simple root of 
d(x), it is easy to determine the nature of the curve in the 
neighborhood of the asymptote. 


464 PLANE GEOMETRY. BOOK IV. 


If ¢,(«) be the first of the coefficients which does not be- 

come zero for «=a, the equation of the curve can be written 
( 1 1 

(= 0) {Ww (0) + @) + + Fara @ f 


i 1 
+5 be @trbn@+ 


W(x), ¥,(#) +++, 1 designating integral polynomials in 2, the 
first of which does not become zero for x = a. 
Now, if one assign to # a neighboring value of a, and if one 


consider one of the very small p roots of the equation in cs 


the sign of each of the parentheses is the same as the sign of 
its first term; moreover, y(x) and ¢$,(a) have respectively 
the same sign as (a) and ¢, (a). 

After this, it is sufficient to consider the binomial equation 


(x —a) ol) +25 bo = 0, 


and to suppose successively « a little less, and then a little 
greater than a. 
(We have already employed an analogous process, § 354, 2.) 
Thus, when the first term of the equation, arranged with 


respect to the increasing powers of 7 has a simple root a, the 


equation can be reduced to this term, and to the first of the 
following terms whose coefficient does not become zero for 
a—a. One can also replace « by a in the factors which do 
not become zero for x= a. 


360. We have, thus far, in equation (1) regarded x as a 
function of —; one can, on the contrary, consider — as a func- 


tion of a. Suppose then that the real root a of the poly- 
nomial ¢)(x) does not reduce ¢,(#) to zero; as # approaches 


a, one value only of 7 approaches zero, and this value is of 


‘necessity real. There exist, therefore, two real branches, 
asymptotic to the same straight line w=a, and one of them 


CHAP. III. ASYMPTOTES. 465 


is given by values of «# less than a, the other by values 
greater than a; these two branches are situated on opposite 
sides of the asymptote. 

In order to determine their position, allow # to vary from 
a—h to a+h, assuming h to be sufficiently small so that, in 
this interval, the equation ¢)(#)=0 has only the root a, and 
the polynomial ¢, (x) does not become zero; it can be supposed, 


moreover, that h, and consequently 7 be sufficiently small in 


absolute value, in order that, when the simultaneous values 
which correspond to a point of one of the infinite branches 


be assigned to # and 7 the value of the polynomial 


1 ae 1 
1 (@)= + 2 (@) + 2 + hn (2) — 
, (x) ; : (2) P bn (#) 7 


has always the sign of its first term ¢; oes but from equa- 
tion (1), the value of this polynomial 8 equal to — qo (a); 
it follows that the two quantities ¢, (a): and — q¢)(x) have 


the same sign, and consequently that : has the same sign 
= $o(*) | 
i (@) 


as When 2 varies from ree to a+h, the de- 








Fig. 226, Fig. 227. 
nominator ¢,(#) preserves the same sign; if a be a simple 


root, or more generally a root of even order, of the equation 
26 


- 466 PLANE GEOMETRY. BOOK Iv. 


d)(x)= 0, the numerator ¢)(#) changes its sign for e=a; 
the value of y changes its sign also, and the two branches 
have opposite directions, one approaching one extremity of 
the asymptote and the other approaching the other extremity 
(Fig. 226), as in case of the hyperbola. When a is a root 
of even order, the numerator preserves the same sign, so also 
does y; the two branches are directed toward the same ex- 
tremity of the asymptote (Fig. 227). 

Suppose now that a reduces the successive polynomials 4, 


do +++ &y-1 to zero. As & approaches a, p values of - approach 


zero; of these values, p — 2q are real for values of a less than 
a, p —2q' for values of @ greater than a; there will be there- 
fore 2» —2q—2q' real branches asymptotic to the straight 
line x= a. 


Examp.e I. — Consider the curve defined by the equation 
atyt + (0? — 4)(y — x)* =0, 
which, expanded, may be written 
(at 4+a2—4)yt—4 x(a? 4) y8 + 6 2?(a?—4) y?—4 93 (42-4) y+ at(x?—4) =0. 
The biquadratic equation 
do(4) = t+ a2?-4=0 


has two real simple roots with contrary signs, 4 


Since these values of x do not reduce ¢1(x) to zero, each of the straight 
lines x =+ a is asymptotic to two real branches situated on the opposite 
sides of the straight line and directed towards its two extremities. 


Exampe.e II.— Consider the curve (x — 1)?y24+4—2?=0. 

The equations ¢o(z) = 0 becomes (% — 1)?= 0. This equation has the 
double root x= 1. When x approaches unity, the two values of y are 
imaginary ; the straight line x = 1 is not therefore asymptotic to a real 
branch. 


CHAP. III. ASYMPTOTES. 467 


ASYMPTOTES WHICH ARE NOT PARALLEL TO THE Y-AXIS. 


361. Let us consider an infinite branch MN of the curve 
(Fig. 228) which has an asymptote CD that is not parallel 
to the y-axis; such an asymptote 
has the equation 


(1) y= ce+d, 


c and d being two unknown con- 
stants which are to be deter- 
mined. Let y and y, be the 
ordinates of the branch of the 
curve and of the straight lne 
which corresponds to the same 
abscissa, and 6 the difference 
y — yy, that is MR; according to the definition, 6 is a function 
of x whose limit is zero when 2 is indefinitely increased. The 
infinite branch of the curve which we consider is therefore 
represented by the equation 








Fig. 228. 


(2) y=y,+8=cra+d+6. 


The equation of a branch of the curve can often be easily 
put under the preceding form, and then the asymptote is found 
as follows. Let, for example, = Fa be the equation, in 
which f(x) and F(#) represent two integral polynomials in a, 
the first of the degree m, the second at most of the degree 
m+1. To each real root of the equation f(#)=0 correspond 
two real infinite branches, asymptotic to the same straight line 
parallel to the y-axis, situated on opposite sides of the straight 
line, and directed toward its opposite extremities or toward 
the same extremity, according as the root a is of an odd or 
even order. There are, moreover, two other infinite branches 
which are obtained by assigning very large positive or nega- 
tive values to a. If the division be effected, one obtains, on 
arranging the equation with respect to decreasing powers of 


468 PLANE GEOMETRY. BOOK IV. 


@, an integral quotient cw +d, which is at most of the first 

degree, whence one has 

(2) 

y=cou+d+ 
S(@)’ 


¢(«) being an integral polynomial of a degree less than m; 
since this last fraction approaching the limit zero as @ is 
indefinitely increased, it follows that the straight line y=cx+d 
is asymptotic to the two branches which we consider. 

We shall cite in addition, as an example, the transcendental 
curve 





yao, 
a 


which has an infinite branch situated in the angle YOX and 
asymptotic to the straight line y = @. 


362. In general, the asymptotes cannot be found so easily. 
Let us return to equation (2). We find 


pee dae 
x £ 
Since d has a finite value, and § approaches zero when a increases 
indefinitely, one has 
(3) c= limit of _ 


t 
v 


The angular coefficient of the asymptote ts equal to the limit 


which the ratio 2 approaches, when x increases without limit. 


The ratio c being the angular coefficient of the straight line 


OM, the relation (3) shows that this straight line approaches 
as a limiting position OZ parallel to the asymptote CD, when 
the point M is removed to infinity on the branch MN. The 
same equation gives d = y — caw — 6, whence 


(4) d = the limit of (y — cz). 


The ordinate at the origin of the asymptote is equal to the limit 
of the difference y — cx, when x increases without limit. 


CHAP. III. ASYMPTOTES. 469 


The quantity y — cx being the ordinate M Q of the curve in- 
tercepted by the straight line OZ parallel to the asymptote, 
relation (4) shows that this ordinate approaches a hmit OB, 
when the point Mis removed to infinity on the branch JLN. 

The two relations (8) and (4) determine the asymptotes 
which are not parallel to the y-axis. 

Suppose that the equation is solved with respect to y, and 
consider a determination of y which gives a real infinite 
branch, when « is increased without limit. We take, for this 


; if this ratio does not approach a finite 
limit, the branch does not have an asymptote. If the ratio 
approach a finite limit ¢, the difference y — cx 1s considered ; 
when this difference does not approach a finite limit, the 
branch does not have an asymptote; if, on the contrary, it 
approach a finite limit d, one will have y—cx=d + 6, where 6 
approaches zero as # increases without limit; therefore the 
straight line y,=ca-+d will be an asymptote of the branch 
under consideration. 


branch, the ratio 


Examp_Le I.— Construct the curve 
x—1 
y= + oy a 2 

referred to rectangular axes of co-ordinates. The x-axis is the axis of the 
curve. When «x varies from o to unity, y 
remains finite ; it is in the first place zero, ye 
then increases and becomes zero again ; 
whence we get the oval OAO (Fig. 229). Y 
As x varies from 1 to 2, y is imaginary. 
When zx becomes greater than 2 by a small 
quantity, y is real and very large ; if, there- 
fore, OB be taken equal to 2, and GG’ be XS 
drawn parallel to OY, this straight line will 
be asymptotic to two branches of the curve. z 

As « increases from 2, y begins to dimin- 
ish, and finally becomes very large, when x 
is large; thus the two branches CND and 
C'N'D! are obtained. When & is negative, 
y is always real; as x varies from 0 to — », 
the numerical value of y varies uniformly D! 
from 0 to o, and so one gets the two Glo" 
branches OF, OL’. Fig. 229. 











AN 


ae 








470 PLANE GEOMETRY. BOOK IV. 


Consider, for example, ND, one of the infinite branches ; this branch 
is found by taking the + sign in the equation, and on supposing that x is 
positive and very large. We have 


ae \" ee 
x \ye— 9? 
the limit of _ is unity. Moreover, one has 


y—2o2(y2=1_1)_,Ve=t—ve=e 
x —2 Vx —2 


and, on multiplying the two terms by the sum of the radicals 








x 


Vee o/eeta Jeo) 





y—-%= 





the limit is }; therefore the straight line y = # + i is an asymptote of the 
branch considered. On dividing both terms of the fraction by x, one sees 
that the difference y —~ is greater than 3, and, consequently, that the 
ordinate of the curve is greater than that of the asymptote ; consequently 
one infers that the branch ND is situated above the asymptote. One 
would discover in a similar manner that the branch OZ’ has the same 
asymptote, and that it lies above the branch. The two branches N’D! 
and OF have as asymptote a straight line which is the symétrique of the 
preceding with respect to the z-axis. 


Exampce II. — Consider the curve yt—y3x+43—2 x%2y=0, constructed 
in § 342. We have expressed the two co-ordinates # and y in terms of the © 


auxiliary variable tat The two branches OA and OB, which are 


found by allowing ¢ to approach zero, do not have an asymptote, since y 
becomes infinite. The two infinite branches OC and OD are obtained by 
making ¢ approach unity. We have, for these branches, tlie limit of 


a= 1; test whether the difference y — x has a limit. The formulas, by 
means of which x and y are expressed in terms of ¢, give 


oo 


y¥—x=(t—l)z= B ’ 





this difference approaches unity, when ¢t approaches 1. Whence it fol- 
lows that the two branches under consideration have the straight line 
y=x-+1asanasymptote. The difference 6 has the value 


pee Ge ee 
t3 : 





when ¢ varies from 1 to + o, the difference is negative and the branch 
OD is situated below the asymptote. The polynomial ¢? + ¢ — 1 has the 


CHAP. III. ASYMPTOTES. 471 


roots t° => awe t= se ; when ¢ varies from } to ¢’/, 6 is nega- 


tive, and the are OF is below the asymptote ; as ¢ varies from ¢! to 1, 6 
becomes positive, and the are EC crosses to the other side of the asymp- 
tote. The other root ¢" gives the point # where the branch OA intersects 
the asymptote. 


363. Let us consider now the case when the equation sup- 
posed algebraic and integral is not solvable with respect to 
the variable y. Collect terms of the same degree; represent 
by (2, y) the ensemble of terms of the highest degree m, by 
w(a, y) the ensemble of terms of the degree m—1, by x(a, ¥) 
the terms of the degree m — 2, ...; the equation may be written 


(8) F(a, ¥) = $Y) FY, Y) + x@,Y) ++ = 9. 


Represent the ratio / by u, and substitute wa for y in equa- 
z 


tion (5); the polynomial ¢(z, y), being homogeneous and of 
the degree m, will contain #” as a common factor in all of its 
terms, and it will follow that o(a, y)=a"(1, u), or, for brevity, 
a"p(u). Similarly the polynomials $(@, y), x(@ Y))-- will be- 
come x” y(u), a”~x(u),... The equation connecting « and u 
will therefore be 


aw) + ay (u) + a Px(u) + +++ =I, 


and, after dividing by 2”, 
(6) CH) +590) + px) +o =O. 


Suppose that a real branch MN (Fig. 228) be asymptotic to 
a straight line CD which is not parallel to the y-axis. When 
the point M is removed to infinity on this branch, w ap- 


ope | 
proaches a finite limit ¢, while — approaches zero. Since the 
a 9 


terms of equation (6), beginning with the second, approach 
zero, it follows that the value w=c annuls the polynomial 
d(u). Thus the angular coefficients of the asymptotes satisfy the 
equation d(u) = 0. 


|S 


Take now y—cv=v, whence u=-=c¢+ <. On substitut- 


8 


472 PLANE GEOMETRY. BOOK Iv. 


ing this value for wu, and developing each term, equation (6) 
becomes 

v c) vw ee 
@) $0) Fo 2 429 24... 


+¥@ +40 - 34} 0. 





i 
+ x(c) ae ay 


Since ¢(c) = 0, if one multiply by a, it takes the form 
(8) [8 +YO]+4A=4+ BSH -- = 
When the point M is removed to infinity on the branch MN, 


v approaches a finite limit d@ whilst : approaches zero. The 
a 


terms of equation (8), beginning with the second, approach 
zero; it follows that the value v=d reduces the first term 
vg'(c) +¥(c) to zero. If ¢ be a simple root of the equation 

(vu) =0, the quantity ¢'(c) being different from zero, one 
obtains the following finite value for d: 


9 eK), 
. 7=— $@ 

364. Conversely, let c be a simple real root of the equation 
¢(u) =; consider the corresponding finite value d given by 
equation (9), and construct the straight line CD, whose equa- 





tion is y=cx+d. Owing to equation (8), when e approaches 
by 


- zero, one value of v, and only one, 
approaches d; this value, necessarily 
real, represents the ordinate MQ, in- 
tercepted between the parallels OF 
and CD; it follows that there are 
two real branches asymptotic to the 
straight line CD, one on the side of 

Fig. 230. the positive and the other on the side 

of the negative a’s (Fig. 230). 

nee that ¢ be a root of the pth order of ¢(u) and does 


not reduce y(u) to zero; according to equation (8), when : 








CHAP. III. ASYMPTOTES. 473 


approaches zero, each value of v becomes infinite; because, 
if one value of v preserves a finite value, the coefficients 
A, B,+-- would remain finite, and equation (8) would reduce to 
W(c)= 0, which contradicts the hypothesis. Owing to equa- 


tion (6), when — approaches zero, p values of w become equal 
a 


to ec; among these p values, p — 2 q are real for very small pos- 


itive values of iS p—2q' for negative values. Draw the straight 
2 


line OE, whose angular coefficient is c. To each real value of 
u there corresponds a straight line OM, making a very small 
angle with OH; the point M, in which this straight line inter- 
sects the parallel to the y-axis with the abscissa a, belongs 


to the curve ; when - approaches zero, the ordinate v = MQ be- 
a 


comes infinitely large in absolute value, the branch described 
by the point M does not have an asymptote, and is similar to 
a branch of a parabola. There corresponds an even number 
2p —2q—2¢' parabolic branches to the direction ec. 

Discuss the. case when c is a double root of the equation 
¢(u) = 0, and annuls y(c). Equation (7) becomes 


a) [SOvsrvortxo|+itgt=0 


When E approaches zero, two values of v approach finite 
z 


limits which are roots of the equation 


(11) FAD + Wu + x()=0. 


If the two roots d and d! of this equation be real and unequal, 


A ae 
one value of v approaches d, when — approaches zero; it 1s 
x 


real and furnishes two real branches asymptotic to the straight 
line y=ca+d. The value of v which approaches d’ fur- 
nishes in a similar manner two real branches asymptotic to 
the straight line y = cx + d', parallel to the first. If the roots 
of equation (11) were equal, one could no longer make the 
preceding deduction; in this case one would introduce a new 
transformation by putting v = d + w. 


474 PLANE GEOMETRY. BOOK IV. 


364. 2. It is easy to find the position of the curve in the 
neighborhood of an asymptote, when the ordinate d of this 
asymptote corresponding to the origin is a simple root of the 


first term of the equation in -. 
; ts 1 
For this purpose reduce the equation in 7, 10 its first term 


and to the first of the terms following whose coefficient does 
not become zero for v=d. One can afterwards replace v by 
d in the factors which do not become zero, for v=d (see. 
§ 359. 2). 


365. Remarks. — We have seen that a simple root ¢ of the 
equation (wv) =0 gives two real branches asymptotic to the 
straight line CD, which has the equation y= ca+d. If a cer- 


tain value be assigned to v, equation (8), in which le is re- 
% 


garded as unknown, will determine the points of intersection 
of the curve and of a straight line y= cx +v parallel to the 
asymptote. Equation (7) being of the degree m with respect 


to , equation (8) is of the degree m—1; whence it follows 
2 


that a parallel to the asymptote intersects the curve at most in 
m—1 points. If the particular value d be assigned to v, the 
equation is depressed to the degree m—2; the asymptote 
intersects the curve at most in m — 2 points. 

Consider next the case when ¢ is a double root and annuls 
y(c); equation (10) being of the degree m — 2 with respect to 


a straight line parallel to y= cx intersects the curve at most 
a 


in m—2 points. If the roots of equation (11) be real and 
unequal, the two asymptotes both intersect the curve at most 
in m — 3 points. 


Exampre. — Let the curve be, ¥# — yx + 23 — 2x2y = 0, constructed 
as in § 342. One has o(u)= ut —w= wW(u—1), Y(u)=1—2u. The 
equation @(w) = 0has a triple root zero and a simple root 1. The simple 
root c= 1, with the corresponding value for d= 1], gives a straight line 
y =x-+1 asymptotic to two real branches. The triple root will furnish 
asymptotes parallel to the x-axis; but it is plain, from the equation of 
the curve, that none of the values of y approach a finite limit, when x 


is increased indefinitely. 


CHAP. III. : ASYMPTOTES. 475 


Equation (8) in this case becomes 


1 1 1 
—1 3 v2 —2v)-+3v3—+ vt—=0. 
(v y+ (80 oot re ae 
If we put v = 1, we obtain an equation of the second degree, 
lier 


which gives the two points # and F in which the asymptote intersects 
the curve (Fig. 216). 


366. It is an easy problem to reduce the investigation of the infinite 
branches of an algebraic curve to the study of the finite branches of an 
algebraic curve of the same degree. Let x and y be the co-ordinates of 
any point M of the figure for the first curve; let the point JZ’ whose 
co-ordinates are x’ and y’ correspond to the point MW, and the co-ordinates 
x' and y' expressed in terms of the co-ordinates of M be 


1 
Bag te 
from which follows, conversely, 
y! 
OS gt Ue 


If the point M describe a straight line Ax + By + C=60, the point M’ 
describes the straight line Cx’ + By’ + A=0; the angular coefficient of 
each of the straight lines is equal to the intercept of the other on the y-axis. 
More generally, if the point MZ describe a curve of the degree m, the point 
M' describes a corresponding curve of the same degree ; to a secant passing 
through two neighboring points of one of the curves corresponds a secant 
passing through two neighboring points of the other curve, and, conse- 
quently, to a tangent there corresponds a tangent. We can assume that 
the first curve is referred to axes in such a manner that the equation 
involves a term in y™; then the infinite branches are obtained by making 


x increase without limit, and all the values of the ratio z approach finite 


limits. Whence, if the point M describe an infinite branch of the first 
curve, since x’ approaches zero and y/’ a finite | 
value c, the point M’ will describe a branch 
intersecting the y-axis at a point A’ whose ee 
ordinate is c (Fig. 281). In this way the 
study of infinite branches of the first curve 
is reduced to the investigation of branches 


of the second curve in the neighborhood of 
points situated on a y-axis. 
Let A’ be a point in which the second Lg 


curve intersects the y-axis; call d@ the an- 
gular coefficient of the tangent at this point ; Fig. 231. 





° 
iad 





476 PLANE GEOMETRY. BOOK IV. 


' _ 
for a point M’ consecutive to A’ one has y _ ry + 6, 6 approaching 





zero with x’; the branch A’M’' of the second curve is therefore repre- 
sented by the equation y¥ =c+ dx’ + dz’ ; to this branch corresponds an 
infinite branch of the first curve whose equation is y=cx+d-+ 4, 6 
approaching zero when z is increased without limit ; to the line y/’ =c+dzx! 
tangent to the second curve corresponds the asymptote y = cx + d of 
the first. We know that an even number of branches having the same 
tangent ($ 353) emanate always from the point A’; the first curve, pos- 
sesses therefore an even number of infinite branches having the same 
asymptote. Since the tangent at A’ is the limit of the tangent at M', it 
follows that the asymptote is the limit of the tangent at the point UM, 
when this point is removed to infinity. 

Suppose, for example, that the point A’ be an ordinary simple point, 
as indicated in Fig. 231; according to the sign of 6 one sees that to the 
branch A’M’ there corresponds an infinite 
branch M situated above the asymptote to 
eee the right, and to the branch A’N’ a second 
oom a eA infinite branch N situated below the asymp- 
tote to the left. If there be an inflection at 
A! (Fig. 232), the two infinite branches 
. [ : would be situated both on the same side of 
> yy the asymptote, one to the right, the other 
to the left; in this case it is said that the 
curve has a point of inflection at infinity. If 
the point A’ be a double point with distinct 
tangents, there will be two parallel asymptotes to each of which there 
corresponds two infinite branches having one of the preceding positions. 
If the point A’ be a cusp, there will be two branches asymptotic to the 
same straight line, but towards the same extremity. ; 

It has been assumed thus far that the tangent at A’ does not coincide 
with the y-axis; if this were the case, to the branches which emanate 
from the point A’ correspond, in the first figure, infinite branches without 
asymptotes. The direction of the tangent at the point has as its limit 
the direction determined by the angular coefficient ¢ ; but it is removed to 
infinity. The name parabolic branches is given to such infinite branches. 
If the point A’ be an ordinary simple point, the two infinite branches 
have the same directions as the branches of the ordinary parabola. If 
A' be a point of inflection, the two branches have opposite directions. 
The curve represented by the equation y® = « presents this arrangement ; 
it is composed of two inflnite branches without asymptotes, and directed, 
one toward the positive «’s, the other toward the negative x’s. 








Fig. 232. 


367. Since to each value of x there correspond at most m real values 
of y, the first curve has at most m infinite branches on the side of the 


CHAP. III. ASYMPTOTES. ATT 


positive x’s, and m infinite branches on the side of the negative x’s; this 
is, moreover, a consequence of the fact that the second curve is inter- 
sected at most in m points by the y-axis. Since the number of tangents 
to the second curve at these points of intersection is at most equal to m, 
the first curve has at most m asymptotes. 

The straight lines drawn from the point A’ transform into straight 
lines parallel to the asymptote. If the point A’ be a simple point (§ 354), 
a secant drawn from this point intersects the curve in m— 1 additional 
points ; therefore every straight line parallel to an asymptote intersects the 
first curve in m—1. The tangent at A’ does not intersect the curve in 
more than m — 2 other points; and, consequently, the asymptote does 
not intersect the first curve in m — 2 points. If A’ be a point of inflection, 
since the tangent intersects the second curve in three points, which are 
coincident with A’ (§ 345), the asymptote will have three points of inter- 
section at infinity ; and, consequently, will not intersect the curve in more 
than m — 3 points. 


368. The transformation which we have made is equivalent to taking 
the perspective of the figure on a plane. Consider two figures situated 
one in the horizontal plane, the other in the vertical plane, intersecting 
the horizontal plane in the line LT (Fig. 283), and in such a way that 








&y 
NY 
----4--\9o 
QR 
~ 








L ees 
aoe 
: 
y' A' 
P M' 


one is the perspective of the other, the eye being placed at the point whose 
projections are 0 and o’. It is evident that, when a point M is removed 
to infinity in the vertical plane, its perspective M' falls upon the straight 
line o’y’ parallel to la ligne de terre (LT); the study of the infinite 
branches of the curve situated in the vertical plane is thus reduced to the 
study of the other curves in the neighborhood of points situated on the 
straight line oy’. 


J 


oO! 
x’ 


Fig. 233. 


len 


478 PLANE GEOMETRY. BOOK IV. 


If one of the curves be referred to the axes ox and oy; the other to the 
axes o/x! and o’y’, one has the formulas of transformation 2! = 2, y!/= os 
in which a and 6 represent the distances ao! and ao. These formulas are 
identical with those which have been used above when one puts a=b=1. 

Let A’ be a point in which the second curve intersects the straight line 
o'y’; the straight line A’B tangent at this point has as perspective the 
straight line A;.1; to the two branches M' and N’, which start from the 
point A’, correspond two infinite branches Mf and N, asymptotic to AA;. 

When the tangent at A’ coincides with the straight line o’y’, as is the 
case at the point C’, the corresponding asymptote is situated at infinity, 
and the two branches P’ and Q! give birth to two infinite parabolic 
branches P and @ ; the straight line drawn from the point 0 to a point of 
either of the branches P and Q approaches the limiting direction 0 C4. 


? 


369. Transcendental curves can intersect their asymptotes in an 
infinitude of points. 


For example, the curve y = “ae oscillates perpetually from one side 


to the other of the straight line OX, to which it is asymptotic, since the 
value of y has the limit zero (Fig. 234). The oscillations have a constant 


y 





ax ae oa 





Wigs 234. 


amplitude which is equal to 7. r 


oscillates perpetually from one side to the other 





in 72 

The curve y = me 

of its asymptote OY; but in this case the amplitude of oscillation 
diminishes continually (Fig. 238). . 

We have seen (§ 360) that, in algebraic curves, if an infinite branch 

have an asymptote, the tangent approaches a limiting position, which is 








Y 
EP ere VAI = 
SO <7 jo we ——— 
Fig. 235, 


the asymptote itself, when the point of contact is removed to infinity on 
‘ the branch of the curve. But this is not true in general in case of trans- 


CHAP. III. ASYMPTOTES. 479 


cendental curves ; thus, in the preceding example the angular coefficient 


of the tangent, 
sin x? 


y' = 2 cosa? — : 
2 


does not approach a limit ; because the second term approaches zero, and 


the first oscillates between — 2 and + 2. 
It can often happen that two infinite branches do not have a rectilinear 


asymptote, and notwithstanding the difference of their ordinates ap- 
proaches zero ; in this case, it is said that the two curves are asymptotic 
to each other; if one of them be a well-known simple curve, it would 


serve to trace the other. Consider the equation y = oe and suppose 


that the degree of the numerator is at least two units greater than that of 
the denominator; the ratio = increasing without limit with z, the 
branches which correspond to very large values of x, positive or negative, 


do not have rectilinear asymptotes. If the > 
division of the numerator by the denomina- _, 
tor be effected, we have 





~=. 
~ 
= 
-<. 


p(%) 
= ax" + bar-1+ ..- +k 472, 
: 7@) 


and the two branches considered are asymp- 
totic to the curve 


y= ax" + bar-14 .. +k, art ie: 
0 x 








When n = 2, the second curve is a parabola. 
For example, the curve 
1 


3 
ye eS 
: x x 





is asymptotic to the parabola y = «? (Fig. 
236). 


480 PLANE GEOMETRY. BOOK IV. 


CHAPTER IV 
CONSTRUCTION OF CURVES IN POLAR CO-ORDINATES. 


370. Polar co-ordinates have been defined in § 3; in this 
system, any point, taken at random in the plane, can be deter- 
mined by a value of w» comprised between 0 and 272, and by a 
positive value of p; however each of the co-ordinates w and p 
may vary from —# to+o. 

We have seen (§ 263) that if one of the foci of the yepeieor 
be taken as pole, its two branches are represented by two 
distinct equations, when we confine ourselves to positive radii 
vectores; moreover, one of the equations is sufficient, if neg- 
ative radii vectores be allowed, on agreeing to measure the 
absolute value of each of them in the direction opposite to that 
indicated by the value of ». We have also seen that this con- 
vention makes it possible to represent the limacon of Pascal 
by a single equation (§ 27). 


371. Spiral of Archimedes.—A point M has a uniform 
motion in the direction G'G on an indefinite straight line 

G'OG which revolves with a 

uniform motion about one of 

its points O. The curve de- 

scribed by the point M is 

the spiral of Archimedes (Fig. 
x 237). 

Take OX for the polar 
axis, the direction which the 
straight line OG has when 
the movable point M passes 

Fig. 231. - through O, and reckon posi- 
tive polar angles in the direction of rotation of the straight 











CHAP. IV. CURVES IN POLAR CO-ORDINATES. 481 


line; let a be the distance which the movable point has 
advanced on the straight line, while it has made a complete 
revolution. If the variable point be considered in any of its 
positions after its passage through O, on calling w the angle 
through which the direction OX has revolved in order to 
coincide with OG, and p the distance of the variable point 


from the point O, one has on putting 52> b, 
Us 


@W 


=5 9 OF p= a 5 -— bo. 


(1) a 


2 fo 


Let us consider the movable point before it passes through 
‘ O; call w, the absolute value of the angle through which it is 
necessary to revolve the direction OX in order to make it 
coincide with OG; the point M being situated on OG', the pro- 
longation of OG, the radius vector should be regarded as nega- 


p 


° Sane 09) 
tive, and one, has ——==.——. 
a 


2a 
measured in a direction opposite to the first as negative, one 
will have w =— w,, and the preceding relation is identical with 
equation (1), which represents an indefinite curve. The values 
of w comprised between 0 and 27, 27 and 47, ---, 0 and 
—2r,-+--, give the successive helices. If we confine ourselves 
to the positive values of p and to the values of w comprised 
between 0 and 27, it would be necessary to employ a particular 
equation in order to represent each of the helices, 


If one regard those angles 


p = bo, p=a-+t ba, >, p=a— ba, p=2a— ba,-. 
The spiral of Archimedes is composed of two parts 
OB,C,D, A,B. and OB FD FE Bs ers, 


symétriques of one another with respect to OY perpendicular 
to the polar axis. Each portion embraces an infinitude of 
helices, and the portions of any straight line drawn through 
the pole and comprised between two consecutive helices all 
have the length a. . 


372. Remark I.— Any point M of the plane can be defined 


by an infinitude of pairs of values of pando, If « represent 
2H 


482 | PLANE GEOMETRY. BOOK IY. 


a positive angle less than 2a which the direction OM makes 

= with the axis OX, and a the distance OM 

Pe (Fig. 238), one can select as co-ordinates of the 

ms = =z point M the pairs of values comprised in the 
Fig. 238. two formulas 





p=+4a4, o=a+2kr, 
p=—a, w= a4+(2k+4+1)z, 


where k is any integral number. If the point M belong to a 
curve defined by an equation f(w, p)= 0, its co-ordinates can 
be discovered merely by inspection of the point; it is necessary, 
in order to obtain them, to follow the trace of the curve. 


373. Remark II.—JIn the formulas of transformation es- 
tablished in Book I., Chapter IV., we have supposed the point 
M determined by a positive radius vector and by a polar angle 
comprised between 0 and 27. Taking in the first place the 
radius vector positive, one can choose as polar angle any of the 

angles which have the direction OM with 

the axis OX (Fig. 239), on agreeing to 
x’ assign to the angle the + sign or — sign, . 

according as the straight line starting 
0 x with the direction OX takes the direc- 
me tion OM by revolving from OX toward 
OY, or in the opposite direction. This 
results in increasing or diminishing the 
angle represented originally by », by a multiple of 27; since 
the sine and the cosine do not change, the formulas remain the 
same. Suppose now that the point M is defined by a negative 
radius vector; the angle w will be one of the angles formed by 
the direction OM' with OX. Since the projection of OM 
on OX is equal to (— p)-cos(z +) or pcos, one has still 
a = pcosw, and similarly y=psinw. Therefore the formulas 
are general. 

When the polar axis OX' does not coincide with OX, the 
position of this axis is defined by the angle « which it makes 
with OX. If w in the formulas w= p cos a, y = p Sin w, which 


* 4 








Fig. 239. 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 483 


are referred to the polar axis OX, be replaced by w'+ «, they 
become # = cos (w'+ a), ¥ = p sin (w! + @). 

Suppose now that the axes of co-ordinates Y 
be oblique and take OX for the polar axis 
(Fig. 240) ; the formulas of transformation are - 
obtained by projecting the two paths OM and 
OPM successively on a perpendicular to OX ° E ¥ 
and on a perpendicular to OY, which gives Fig. 240. 


pon p sin (6 _ w) — p sin o 
sin 0 : sin 6 








374. Remark III.—In the case where the entire curve is 
obtained by varying w from 0 to 27, one perceives the sym- 
metry of the curve with respect to the polar axis OX, if the 
values « and 27 —« assigned to w give the same value for p, 
or if the values « and 7—a give values for p, equal and con- 
trary in signs. Similarly, the symmetry of the curve with 
respect to the perpendicular OY is seen, if the angles « and 
a —« give the same value for p, or the angles # and 27—« 
give values for p equal and contrary in sign. Finally, the 
symmetry of the curve with respect to the pole may be seen, 
if the angles « and 7+ give the same value for p, or if to 
the same angle « there correspond two values of p, equal and 
contrary in signs. 

But if, in order to obtain the entire curve it be necessary 
to give w values greater than 27, the symmetry of the curve 
may be discovered in another manner. For example, if it be 
necessary to vary w from 0 to 47, the symmetry with respect 
to the polar axis will exist, if the angles «# and 27—a, or 
a and 47— a, give equal values to p, and moreover if the 
angles a and r—a, or a and 37 —a, give values to p, equal 
and contrary in sign. If the limits of w be farther extended, 
the number of comparisons is increased. 


Consider, for example, the curve defined by the equation p = cos = 


If w be increased two times 27, the direction of the radius vector re- 
mains the same; moreover, if ° be increased by multiples of 27, p takes 


484 PLANE GEOMETRY. BOOK IV. 


the same value and one finds a point previously known; it is sufficient 
therefore that w vary from 0 to47. If 
Y w be increased by 27, the radius vector 


returns to the same direction; but : 


being increased only by 7z, p takes the 
same numerical value with a change in 








ae : a sign; it follows that the portion of the 
locus given by the values of w com- 

< prised between 2 r and 47m is the symé- 

ti trique with respect to the pole of the 

Fig. U1. portion given by the values of w com- 


prised between 0 and 27; in other 
words, the pole is the center of the curve. Forw=a and w=27—a, 
the values of p are equal with contrary signs ; here one has two points 
situated symmetrically with respect to OY perpendicular to the polar 
axes (Fig. 241); this straight line OY is an axis of the curve. The 
variable w varying from 0 to x, as p diminishes from 1 to 0, one obtains 
the arc ABO tangent to the straight line OX at O. The values of w 
comprised between m and 27 give the arc OBA’, the symétrique of the 
first with respect to OY, and the values w comprised between the values 
2m” and 47, the curve A’B’OB'A, the symétrique of ABOBA!' with 
respect to the pole. The curve is closed and consists ef four equal 
ares. The polar axis is also an axis of the curve, which is at once evident 
on noticing that one gets equal values of p for the values a and 47—a 
of w. 


TANGENT. 


375. Let M be a point of a curve referred to polay co-ordi- 
nates. Consider the tangent MT at a point M (Fig. 242), and 
the prolongation MA of the radius 
vector in the direction OM; let V 
be the angle through which it is 
necessary to revolve the prolonga- 
tion MA of the radius vector about 
the point M, in the positive direc- 
tion, in order to make it coincide 
with the tangent. In order to de- 
termine V, call p and w the co-ordi- 
nates of the point of contact MM, 
p+ Ap, w+ Ao those of a neighbor- 
ing point M' of M, and U the angle formed by MA and the 








Fig. 242. 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 485 


chord MM’. When the point M' approaches M indefinitely, 
the chord MM’ approaches the tangent MT, and the angle U 
approaches V. i ee 
cae UA iva, OM _ sin OM _sin(U—dAo) 

The triangle OMM' gives OM’ sin OMM' ie aT Dy 
Since Ap approaches zero when the point M' approaches M, 
one can suppose it to be sufficiently small that p and p+ Ap 


have the same sign: therefore one has in magnitude and in sign 








Me: p 
OM' p+Ap 





and the equation above gives 


p _ sin(U— Ae) 

















: Aw 

a sin (U — Aw) on yy : sin (U — Aw) 
whence Ap sinU—sin(U—Ao) |. Aw cos (U — 4 Aw) 

AG | Aw 


ree oo: 
When Aw approaches 0, the ratio = approaches the derivative 
W 
p' of p with respect to w, and U approaches V. Therefore one 


has 
fee 
tang V=—; 
p 


Remark.— When the radius vector becomes zero for a par- 
ticular value w) of w, one has a branch of 
the curve OC passing through the pole 
(Fig. 243), and the tangent to this branch 
at the pole is the straight line OA de- 
termined by the angle w. In fact if a 
neighboring point M be taken and if the 
secant OM be revolved so that the point Fig. 213, 
M approaches the point O, p becomes zero 
and the secant approaches OA as a limiting position. 








486 PLANE GEOMETRY. BOOK Iv. 


376. Equation of the tangent. The equation of the secant 
MM", (Fig. 244) is (§ 83, 2) 








1 

_ COS w $1n w 

p 

si 

— COS w, SIN w =o, 
Pi 

a al 

Pi Pi 


where p; and w, are the co-ordinates of the point M, and re 
Pi 


Aw, the increments which E and w, take when one passes from 
1 

MY the point M, to the neighboring point MW‘. 

: On subtracting, in the preceding deter- 

\ minant, the elements of the second row 

Nv Mr from those of the third and dividing all 

~ Pal the elements of the third row of the new 

4 * determinant by Aw, one obtains, for the 

secant, an equation which gives, when Aa, 














: approaches zero, the equation of the tan- 
ele ans gent at M,. 

i 
~ COSw sinw 
p 
1 
~- COBG@), Sina, | — Y; 
Pl 
a ! 

() —sinw, Cosa, 

1 





or, on developing, 


i eres ae 
1 1 


Pp 


Sub-tangent. The sub-tangent S, is the radius vector of the 


tangent which corresponds to the polar angle w» = w, + a One 
has therefore 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 487 


If, in Fig. 244, w, be the angle «O.M,, the sub-tangent is nega- 
tive and equal to — OT. 

Sub-normal. The sub-normal S,, is the radius vector of the 
normal at the point M, which corresponds to the polar angle 
2. 
2 

In Fig. 244, the sub-normal is positive and equal to + ON. 

Whatever be the arrangement of the figure, the triangle 
TM,N is right-angled at M, and the point O is the foot of the 
perpendicular dropped from the point MZ, upon the hypotenuse 
TN: this point O will lie therefore always between T' and JN, 
and, consequently, the sub-tangent and the sub-normal will have 
opposite signs: since the absolute value of their product is 


OF, ON = OM, = pr’ 


o = + 


it follows that S,-S, =— pr 
and, consequently, from the value of S,, 
S act pi’ 


Equation of the normal. The normal is a straight line pass- 
ing through the point Jf with the co-ordinates (p;, ), and 
through the point N, the extremity of the sub-normal with the 


co-ordinates { p'1, + The equation of this straight line 


is therefore 








1 ; 
_ cosSw = sinw 
p 
1 
_ cose, sinw, |=9, 
Pl 
us , 
rs OR SIN w, COS @, 
Pi 
- 1 1 . 
or = = — cos (w — w) + —- sin (w — o). 
Pei. Pp} 


377. Examp.e I.— The spiral of Archimedes. Since the equation 
of this curve is p = bw(§ 371), it follows that p’ = b, whence 


bw 


=—_— =. @& 


tanV=f 
p b 


488 PLANE GEOMETRY. BOOK Iv. 


If the point M begin with the pole and advance along the curve, the 
angle V, at first zero, increases constantly and approaches a right angle. 
The sub-normal is constant and equal to b. 


Examp.e II.— Logarithmic spiral. * The curve whose polar equation 
is p = ae™, a being a given length and ma given number, is called a loga- 
rithmic spiral. Suppose that the constant m is positive: if wincrease from 
zero to infinity, p will increase constantly from a to infinity, which gives 
an infinite branch ABC... consisting of an infinitude of circumvolutions 
about the pole (Fig. 245). If w vary from 
0 to — », p constantly dimin shes and 
approaches 0; a second infin:te branch 
AB'C'... is found which makes an infini- 
tude of circumvolutions about the pole, 
constantly approaching this point. If 
the constant m be negative, the positive 
values of w would give the branch which 
approaches the pole, and the negative 
; values the branch which recedes from 
Fig. 245. it. In this case one has p’=mae™ = mp, 








hence tan V = i. Whence it follows that the tangent to the curve makes 
m 


a constant angle with the radius vector. 


378. Exampce III. — Epicycloid. When a circle rolls, without slid- 
ing, upon a fixed circle, a point of the rolling circle describes in the plane 
a curve which is called an epicycloid. 

Let us consider the case when the two circles are equal. Let ( be the 
fixed circle, C’ the initial 
position of the rolling 
circle, and a the radius ; 
suppose that the point of 
contact be A, which, by 
the motion of the circle 
C’, generates the epicy- 
cloid (Fig. 246). When 

x the rolling circle has 
taken the position C", 
the point A is at M, and 
the two arcs EA, EM 
are equal. Take the 
point A as pole, and 

Fig. 246. CA prolonged as polar 
axis. The straight line 

AM, perpendicular to EN the common tangent to the two circles, 











CHAP. IV. CURVES IN POLAR CO-ORDINATES. 489 


is parallel to CH; the angle AHN is a half of the angle ACE, and, 
consequently, a half of w; the right triangle ANE gives AN= AE sine; 


but AH =2a sin; one has, therefore, 
p=4a sin? = 2a(1 — cosw). 


This curve is a particular case of the limacon of Pascal (§ 26). Here 
one has p! = 2a sinw, whence tan V = tans, and, consequently, V=<. 
It is easy to see that the normal to the epicycloid at any point MW passes 


through the point of contact # of the rolling circle with the fixed circle ; 
because the angle MEN being equal to = and, consequently, to V, the | 


straight line #M is perpendicular to MT. 


379. Exampie IV.— Construct the curve p=4+cos5w. The radius 
vector p is always comprised between 3 and 5; construct with radii 3 and 
5 two circles about the pole as center; the curve will be wholly situated 
between these two circumferences 
(Fig. 247). When w varies from 


o to 7 p diminishes from 5 to 8, 


which gives the arc AB. As w 
varies from 7 to =, p increases 


from 3 to 5, which gives the arc 
BA' the symétrique of the first 
with respect to the straight line 
OB. The angle 5w has varied 


from 0 to 27. If w vary from 


| - to af, the angle 5 w will vary 








from 27 to 47, and the same values Fig. 247. 
of p will be reproduced in the same 
order ; a second arc A’B’ A" is found equal to the first, then a third, and 


soon. With the fifth arc one will arrive at the point of departure. By 


p At the points A 


constructing the derivative one finds tan V= — -——__. 
5sin iw 


and B one has sin5w = 0, and, consequently, Vas 

380. Exampte V.—The extremities of a straight line of constant 
length slides on two straight lines OX, OY which are perpendicular to 
each other; from a fixed point Jon the bisector of the angle XOY one 
draws a straight line perpendicular to the variable straight line ; find the 
locus of the foot of the perpendicular (Fig. 248). It is evident that the 


490 PLANE GEOMETRY. BOOK IV. 


locus will be symmetrical with respect to the straight line OF. Consider 
first the variable straight line in the position PQ perpendicular to the bi- 
sector; a point A of the locus is determined. Make the straight line move 
so that the extremity Q descends along the y-axis ; in some position P!Q! 


a & 























ad 


, 





Fig. 248. 


will pass through the point 7, which belongs to the locus ; whence one has 
the arc AEFI, whose tangent at J is perpendicular to P'Q'. The extremity 
Q' continuing its descent will coincide with OX, and one has the arc 
IFC which passes through the point C, the foot of the perpendicular let 
fall from Ito OX. As the extremity Q slides along OY’, the curve passes 
below OX; the straight line will arrive in a certain position P/Q" such 
that the angle IP!'Q" is right, which gives the point P!’ of the locus ; 
whence we have the are CGP". If the extremity Q” continue its descent, 
the curve will return above the z-axis, and the straight line will finally 
assume the position P!/’Q!'’, which prolonged passes through J; we get 
‘then the arc PI whose tangent at J is perpendicular to PUG At 
the extremity P’ continue to approach the point O, the straight line 
will ultimately coincide with the y-axis and we obtain the arc JHD, 
which passes through the point D, the foot of the perpendicular 
dropped from the point 7 upon the y-axis, If the extremity P'"' slide 
along OX’, the straight line will assume a position P*Q such that the 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 491 


angle TPQ is right; the point P# belongs to the locus, and we obtain 
the arc DP. Finally the straight line, in some position P’Q’, becomes 
perpendicular to the bisector OB of the angle X’OY’, which gives the 
arc PB. If returning to the initial position PQ, the motion of the 
straight line be reversed till the final position P’Q” is reached, it is clear 
that a curve the symétrique of the first with respect to the straight line 
AB will be found. 

Take as pole the point J and as the polar axis the bisector BA; call c 
the distance OJ and 2 a the length of the variable straight line PQ; the 
straight line which joins the point O to the middle of the hypotenuse PQ 
of the right triangle POQ is equal to a; the angle “which this straight 
line makes with the perpendicular h dropped from the point O upon the 
hypotenuse is equal to 2w; one has moreover h =ccosw +p; whence 
follows the equation of the curve p = @ cos 2 w — € COS w. 


CONVEXITY AND CONCAVITY. 
381. Consider on an are of the curve a point M, whose 
co-ordinates are wy and py; the tangent at this point will be 


represented by the equation 7 = , if qg be the length 





qd 
Cos (w — f) 
of the perpendicular let fall from the pole upon the tangent 
and 8 be the angle which the perpendicular makes with the 
polar axis (§ 82). The position of the curve with respect to 
the tangent, in the neighborhood of the point M@, depends upon 
the sign of the difference r—p of the radii vectores for the same 


ieee _ 
value of w, or of the difference — — 7) We assume that the radii 


vectores are positive. Let z be this last difference; the value 
of z is evidently zero at the point MW; its first derivative 


‘=()-G)-G 


is also zero; because one has, at the point M (§ 375), 








! ! t : : 
() =—-4=-— sin (wo — B) — COS V,; ¢= Po S10. V. 
0 


The second derivative, 


1\"" cos (w — B) =)" 1 
(ae ioe ae ai = 
: G) Peg p) TF 





492 _ PLANE GEOMETRY. BOOK IV. 


has at the point M a value equal to that of the expression 
" 

a (*) . On repeating here the reasoning of § 344, it can be 

PP 


easily verified that, if this quantity be positive, the difference 
zis also positive in the neighborhood of the point M, and con- 
sequently the curve is situated on the same side of the tangent 
as the pole, and that, if on the contrary this quantity be nega- 
tive, the difference z is negative, and the curve will lie on the 
other side of the tangent. 

If the radius vector p be negative, one sees by similar con- 
sideration that the position of the curve with respect to the 
tangent is on the same side as the pole when the quantity 


" 
oat (5) is negative and on the other side if this quantity be 
p 


positive. 
‘In general, therefore, it may be said that at a point M of a 
curve it turns its convexity or its concavity toward the pole accord- 


ing as 
ae?) J 
eas eames Bee 
pLe \p p 


is negative or positive at this point. 
There is a point of inflection at the point of the curve where 


' 
the quantity a () changes its sign. 
p 


ft 
v 


ASYMPTOTES. 


382. Consider an infinite branch asymptotic to the straight 
line CD (Fig. 249); if the point WM of the curve be joined to 
the pole, and if the point M be removed 
toward infinity on the curve, the radius 
vector OM will have for its limit a line” 
OL parallel to the asymptote. Thus, 
when the radius vector p becomes infinite 
for a particular value « of wo, if the 
branch thus determined have an asymp- 
tote, this asymptote is parallel to the direc- 
tion determined by the angle a which 
makes p infinite. 








CHAP. IV. CURVES IN POLAR CO-ORDINATES. 493 


To find the distance OC of the asymptote from the straight 
line OL, draw from the point M, MX perpendicular to OL; 
the triangle MOK gives 

MK = OM sin KOM = + psin(a — a). | 

If the product + psin(@—w) do not have a finite limit, the 
infinite branch does not have an asymptote. If, on the con- 
trary, this product approach a finite limit, the branch of the 
curve does have an asymptote, the straight line CD, situated a 
distance OC, equal to this lmit, from the line OL; because 
if the distance M/~ have the lhmit OC, the distance MH will 
have the limit zero. 

A second demonstration. Suppose that the infinite branch 
is referred to two rectangular axes drawn through the pole, 
the y'-axis in the direction a, and the «'-axis in the direction 


a—". If Ow' be taken for a new polar axis, it follows that 
wo! = wo — (« — 5) and the abscissa w' of the point M will be, in 
every case (§ 373), 

2 cose! = pcos (a — @ =) = p Sin (® — w). 


Qne is thus led to seek an asymptote parallel to the y-axis. 
The abscissa gq of the asymptote is 
the limit of a’, when the point M is ,,, | 
removed to infinity on the branch of 
the curve; one has, therefore, 


q=limp sin(a@ — ). 


The absolute value of qg gives the 
distance of the asymptote from the 
straight line Oy’; the sign shows on 
which side it is situated. se Swe 





383. Exampie VI.— Zhe Hyperbola. The polar equation of this curve 


is (§ 263) p= Tee in which e is greater than1. Leta be the angle 


Pel hae 1 : : 
whose cosine is = 33 when w increases from 0 to a, p increases from 





. to oo, and one has the infinite branch AH; when w varies from a to 
e 


1, p becomes negative and varies from — o to -#i, which gives the 


fr vik RARE 
OF THE ’ 
UNIVERSITY } 


494 PLANE GEOMETRY. BOOK IV. 


infinite branch Z’A’. The values of w comprised between 7m and 27 give 
two branches which are the symétriques of the preceding with respect to 
the polar axis. 
The distance MK of a point of one of the branches AF, A!H!' from the 
line OL is 


ur —2 in (4 —&) _ p sin (a —#) 
1+ecosw 1 
é (= + cos) 





_ psin (a — w) 
~ €(cosw — cosa) 








Substituting a product for the differ- 
a—-w 


2 


ence coSw— cosa, and 2sin 




















cos = : © for sin (a — w), and suppress- 
| ing the common factor sin “—®, one 
Fig. 251. has 
pcos * - = 
nh 
esin2 +2 
2 


This distance has the limit OC = tee thus is obtained the asymptote 
CD. The asymptote of the other two branches is situated symmetrically 
with respect to the polar axis. 


The difference between MK and its limit is 


2 sin a cos *#—% — 2sin2=° 
=? x ; 


2sin asin sr 


a—w ) 
5—? cake 
é sinS sina 





cos 











if the product 2 sin acos*—* be replaced by the sum 





8a 


° — oe (oo) 
sin-———— + sin + 


2 





’ 


-_oa—w : 
oe 





a+w 
2 





it becomes 6=~-: tea ; 
2 ain a sin —— 


if the numerator be transformed into a product, one has finally 


a-—-wWw @.— & 


COS a p sin 





p sin 





6= 





a+ _ sina sin 





: : w+a 
ésin a sin = 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 495 


When w varies from 0 to a, the difference 6 is negative; thus the 
branch AF is comprised between the parallels OL and CD. But, when 
w varies from a to 7, the difference 6 is positive, and the branch E'A’ is 
situated without the parallels. 


384. Exameprte VII.— Oblique Strophoid. In the construction of 
the right strophoid, such as has been given in § 23, we suppose the straight 
lines OX and OY perpendicular 
to each other; suppose now that 
these straight lines include an 
angle @ (Fig. 252); through the 
fixed point A, situated on one of 
them, draw any secant AD, on 
which take, beginning with the 
point D, the lengths DM and DN 
equal to DO, and find the locus of 
the points M and N. When the 
secant revolves in the obtuse angle 
AOY till it becomes parallel to 
OY, the point M describes the 
arc OMB, ending at the point B 
on OB perpendicular to OY; the 
point N describes the _ infinite 
branch ON. If a distance OG be 
taken equal to OA, and a straight 
line H'H be drawn through the 
point G parallel to OY, one ob- 
tains the asymptote of the branch Fig. 252. 

ON; because WF is equal to AM, 
having the limit AB, the distance of the point N from the straight line 
has the limit zero. 

Allow the secant to revolve in the adjacent angle YOY’; the perpen- 
dicular erected at the mid-point of OA intersects OY' at a point C, so 
that one has CA = CO; when the secant occupies the position AC, one 
of the points arrives at A and the other at #; thus, the secant revolving 
from the position AO to AC, one obtains the arc OM'A tangent to the_ 
straight line AC at A, and the arc OH. As the secant continues its 
motion, the straight line OD!’ becomes greater than AD! or D'F’, and 
the point N’ is situated beyond the asymptote ; one obtains the infinite 
branch #N’ and the arc AM’ B which is a continuation of the arc OMB. 
It is easy to see that the tangents at the point O are the bisectors of the 
angles formed by the straight lines OX and OY. 

If the point O be taken as pole, the straight line OXY as the ae 
axis, and if one call the distance OA, a, and the angle YOY, 6, the 








496 PLANE GEOMETRY. BOOK IV. 


angles DOM and DMO being equal to @ — w, the angle OAD to 6 — 2a, 
the triangle OMA gives the relation, 


_ asin (6 — 2) 
© ee Paine as} 





By aid of the equation, one may easily verify the properties which we 
have deduced from the geometric definition of the curve. 

If the straight line OY be taken as polar axis, the equation of the 
curve becomes _ 


(2) 





_asin(2 +6) 
ie sin w 


385. Exampete VIII. — To Jind the locus of the points of contact 
of tangents drawn from a given point P to the various curves of the 
second degree whose foci are two fixed points F and F’'. 

Take FF’ as x-axis (Fig. 253), and the perpendicular erected to this 
straight line at its mid-point as the y-axis; the general equation of 

conics whose foci are F' and F” is 


so ee 
@) at Sa 





where c designates the distance OF, and 
a is a variable parameter; when a is 
greater than c, the curve is an ellipse; 
when a is less than c, it is a hyperbola. 
Let a and B be the co-ordinates of the given 
point P; the equation of the chord of con- 
tact of the tangents drawn from the point 
P to the conic (1) is ee 











Fig. 253. (2) ee 


The equation of the locus is found by eliminating the parameter a between 
equations (1) and (2). If these equations be subtracted member from 
member, one obtains 





_ —A(y— by) 
ay? — at — By” 








whence a? = 


by substituting in equation (2), the equation of the locus is found 


(3) (x? + y? — ax — By) (Bu — ay) + 2 (x —a)(y — 8)=0. 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 497 


The locus is of the third degree, it passes through the given point P, 
through the foci F and F’, and through the projections of the point P 
upon the straight lines OX and OY. 

If the axes be transferred parallel to themselves to the point P, the 
equation of the locus becomes 


(4) (x? + y? + ax + By) (Be — ay) -+ xy = 0. 


Transforming the equation to polar co-ordinates, taking the point P as 
pole, and the line P_X’ as polar axis, one has 


_ (c? + 6? — a?) sin 2 + 2a8 cos2 w 
ae 2 (asin w — B COS w) 





(5) 


By introducing auxiliary angles ¢ and ¢,, determined by the formulas 


2 aps 
c2 + B2 — a? 





tan ¢ =f, tan ¢4 = 


this equation takes the form 





_ dsin (20 + 41) 
) ~ gin (@ = rs) 


where the letter d designates the quantity 








Fea + B2 — a2)2 + 4 42g? 
~ 3 a? + # 


If the polar axis be revolved through the angle ¢, equation (6) becomes 
identical with equation (2) of the oblique strophoid (§ 384). The angle 
¢ being equal to POA, the asymptote is parallel to the straight line OP. 
Among the confocal curves considered are a hyperbola and an ellipse 
which pass through the point P; one infers that the point P belongs to 
the locus and that the tangents at this point are the bisectors PQ, PQ! of 
the angles formed by the straight lines PF and PF’. The strophoid is 
determined by two straight lines PH, PJ, and a point J on one of them. 
We know the straight line PH; the straight line PZ is determined by the 
fact that the tangent P@ is the bisector of the angle HPI; the point I 
is determined by means of one of the points of the locus, for example, 
by the point A; the straight line ZA should be such that KA = KP; the 
point K is therefore the mid-point of the diagonal OP. 

The preceding curve is thus the locus of the feet of the normals drawn 
from the point P to the curves of the second degree, whose foci are the 
points F and F’; because in case of an ellipse and a hyperbola which in- 


tersect at right angles, the tangent to one of them is normal to the other. 
91 


498 PLANE GEOMETRY. BOOK IV. 


386. Exampre IX. — Construct the curve given by the equation 


The value of p becomes zero for w = 0, and becomes infinite for w = 3; 
draw through the pole the straight line Z’Z, which makes the angle } with 
the polar axis (Fig. 254). When 
w varies from 0 to 3, p is nega- 
tive, and varies from 0 to —o@; 
one obtains an infinite branch 
OA'B', tangent to the polar axis 
and comprised within the angle 
X'OL'. When w exceeds } and 
~ increases from } to o0, p becomes 
positive and decreases from o to 
a; one obtains an infinite branch 
Fig. 254, BA, which makes an infinity of 
circumvolutions about the circle 
described about the pole as center, with a radius a, continually approach- 
ing the circle. When w varies from 0 to — #, p remains positive and 
increases from 0 to a, which gives the branch OZ’ within the circle. This 
branch makes an infinity of circumvolutions continually approaching the 
circle. 
Consider the infinite branches A’B’, AB; the abscissa of a point of 
one of them, with respect to the axes according to § 382, is 








sin (w — 3 
G (w — 4) ; 


5 ’ 


x'=psin(4—w)=-a 
es 3 


its limit is oe ; the two branches have therefore as asymptote CD, whose 


intercept is g = a 


If one put w = } + w’, one has 
8 = 2! —q=—* [wl — (14 2w/) sinw']. 
2 w! 


The quantity placed within the parentheses becomes zero for w!=0 ; its first 
derivative also becomes zero, but its second derivative is negative ; if w 
increase from zero, one infers that the first derivative commences by 
decreasing and, consequently, is negative and the same is true of the 
quantity itself ; thus the difference 6 is negative for the positive values of 
w! made sufficiently small. One perceives in the same manner that the 
difference 6 is positive for very small values of w’, which is evident from 
the formula ; hence the two infinite branches with respect to the asymp- 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 499 


tote have the position indicated in the figure. It is evident, moreover, 
that this asymptote is intersected by the curve in an infinite number of 
points. 


387. Example X.— Construct the curve 
pees WA ee 
sin w 


Since the radius vector takes the same value when w + 27 is substituted 
for w, it is sufficient to vary w from 0 to 27. ‘That the radius vector be 
real, it is necessary that the quantity under the radical be positive. The 


numerator changes its sign for the values m 22 of w and the denominator 
for the values 0 and 7; on arranging these angles in order of magnitude 
0, 7, ot 

A & : 

it is evident that the quantity under the radical is negative from 0 to _ 


——~, 8, 2m, 


positive from ™ to on, negative from om to m, positive from mr to 27; 
, eee i 
the whole curve is therefore obtained by varying w from 6 to a and 


from 7 to27. We notice, moreover, that the supplementary values of w 
reproducing the same values of p, Yr 
the curve is symmetrical with re- 
spect to OY perpendicular to the 
polar axis OX (Fig. 255). 

About the pole as center, de- 
scribe a circle with a unit radius: 
this circle will bisect each of the 
chords which pass through the 





. T 
center; when w varies from rr to 


7 the value of the radical in- 








creases from 0 to 1, which furnishes 
the two arcs AB and AO, either of Fig. 255. 

which is the continuation of the other, tangent to the straight line OA; 
the are AO is tangent to the straight line OY at the point O. By varying 


w from 5 to 27, we obtain the are BA'O, the symétrique of BAO, with 
respect to the straight line OY. 

When w varies from 7 to “2, the value of the radical decreases from oo 
to V3, which gives the two infinite branches EF and CD; by varying w 
from - to 27, the two branches FE! and DOC’, the symétriques of the 


preceding are determined ; these infinite branches are asymptotic to the 
polar axis. 


500 PLANE GEOMETRY. BOOK IV. 


388. Remarx.—It has been shown (§ 382) that if the 
radius vector becomes infinite for a finite value @ of w, the 
position of the asymptote is determined by the formula 


q= lim p sin (a = w). 


The first factor becomes infinite when the second factor ap- 
proaches zero. In the preceding examples, one can without 
difficulty find what the product must be for values of  con- 
secutive to a. In case there is much difficulty, one may employ 
another method. 

One may deduce from the preceding formula 


a 


o— O p 
sin(wo—a@) w—@ 





ae 
qd 


The limit of the first ratio is equal to unity. If : be regarded 


a function of w, the numerator of the second ratio is the incre- 
ment which this function receives when the polar angle varies 
from the value « to the value’w; the limit of the second frac- 


tion is therefore the derivative of a and we have 


“--() for wo = @. 
qd p f 


As a rule the value of p assumes the form p =o, the 
W 


denominator becoming zero for »=«, while the numerator 
preserves a finite value different from zero. We find for the 


derivative 
(1). F(o) (0) —f()E). 
F?(w) 








! 
which reduces to ao for w=a. Thus is obtained the formula 


C3) 
Pea). 


which is very convenient in practice. 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 501 


EXERCISES. 


1. Find the locus of the vertex of a variable parabola which has a fixed 
focus and which touches a conic which possesses the same focus (limagon 
of Pascal). 

2. The vertex O of a variable triangle AOB is fixed, the vertex B 
slides on a fixed straight line OX; find the locus described by the point 
of intersection of AB with the perpendicular erected to the side OA at 
the point O. - 

3. A fixed point O and a fixed straight line OP are given; find the 
locus of the vertex M of a variable triangle MON which fulfills the 
following conditions: the side ON is constant and equal to a, the side 
MN=av?2, finally the angles satisfy the relation cos (MON—2 OMN)= 
cos MOP (lemniscate); show that the tangent at any point M of the 
locus passes through the center of. the circle circumscribed about the 
triangle which has given this point. 


4. A triangle OAB right-angled at O is given, a variable conic 
is circumscribed about this triangle so that the normals at the three 
points A, O, B pass through a common point; find the locus of this 
point. 

5. Find the locus of the foci of the parabolas which have a common 
chord and a common tangent parallel to this chord. 

6. Find the locus of the vertices or of the foci of an equilateral hyper- 
bola, whose center is fixed and which passes through a fixed point. 


7. Find the locus of the center of a given equilateral hyperbola 
required to pass through two fixed points. 

8. One is given aright angle YOX, and a fixed point Pon the bisector 
of this angle ; find the locus of the foot of the perpendicular drawn from 
the point P to a variable secant which cuts off in the angle a triangle 
whose area is constant. 

9. At any point M of a parabola a normal is drawn which is pro- 
longed till it intersects the axis in the point N; find the locus of the 
point of intersection of the tangent to the curve at M with the perpen- 
dicular to the axis at the point N. 

10. Substitute in the preceding problem the hyperbola for the parabola ; 
take the point Mf on one of the axes of the curve ; find the locus. 

11. A parabola revolves about its focus; find the locus of the points 
of contact of the tangents drawn parallel to a given straight line. 

12. Find the locus of the mid-point of a chord normal to a given 
hyperbola. 

18. An ellipse is given; the center of a circle with constant radius 
travels on a diameter of the ellipse; find the locus of the points of inter- 
section of the secants common to the circle and the ellipse. 


502 PLANE GEOMETRY. BOOK IV. 


14. An equilateral hyperbola is given; the center of a circle, which 
always passes through the center of the hyperbola, touches an asymptote ; 
find the locus of the points of intersection of the common secants. 

15. Find the locus of the points of contact of tangents drawn from 
a given point to the circles which touch a given straight line at a given 
point. 

16. Find the locus of the points of contact of tangents drawn from 
a given point to the circles which pass through a given point and touch a 
given straight line. 


17. Find the locus of the mid-points of the chords inscribed in a given 
hyperbola and tangent to a circle concentric to the hyperbola. 
18. Study the loci represented by the equations 
yt —at+2ax7y=0, at*+ yt—2 ay —2b2a2y = 0, 
(x? + y?)?2 — 6 axy? —2az3 4+ 2a92?=0, y= at b(x— 0)”, 
et+yi—3a3—422-—0, z3yBty—2=0, 
yt — at — 2 bry? —2a2z3 = 0, ie ies ers: 
yt — xt — 96 ay? +. 100 a22? = 0, 223 — y3 + (y—a)2?= 
19. Construct the curves represented by the equations 
2siny—sinx=0, sinxsiny=} 


ev : 
oes Y= x7, yY=wuX. 


20. Construct the curves represented by the equations 


sin w lsin 3w 


ao eg = I 
2cosw—1 7 a TO 


p2cosw —2psinw + costw=0, p*cosw —4psinw —tanw = 0, 


1 ee 
ee 2 COS w_ Pa 
sin w 


21. Construct the curve defined by the equation 
y=2sin ie 
x 


This curve, which passes through the origin, does not have a tangent at 
this point. 

22. Find the envelope of straight lines such that the segments inter- 
cepted on them by two fixed conics have the same mid-point. Discuss 
the case when the two conics have in common a conjugate diameter with 
the same direction as the chords. 

23. A variable conic circumscribed about a triangle ABC is such that 
the normals to the conic at the points A, B, C are concurrent in a point 
M. Find the geometric locus of the point M. 


CHAP. IV. CURVES IN POLAR CO-ORDINATES. 503 


94. A variable conic inscribed in a triangle ABC is such that the 
normals to the conic at the points of contact are concurrent in a point 
M. Find the geometric locus of the point NV. (This locus is the same as 
the preceding. ) 

95. Find the locus of the points of intersection of tangents common to 
a fixed conic § and to a variable conic passing through four fixed points 
& 2, C, 2. : 

(a) The points A, B, C, D being arbitrary, the locus is a curve of the 
sixth degree, having as double points the centers of the pairs of straight 
lines which pass through the four points. 

(b) If two points, A and B for example, be on the fixed conic S, the 
locus resolves itself into a conic and a curve of the fourth order. 

(c) If the three points A, B, C be on S, the locus is composed of three 
conics. : 

(d) If the four points A, B, C, D be on S, the locus is composed of 
straight lines. 

96. Find the locus of points of contact of tangents common to a fixed 
conic § and to a variable conic passing through four fixed points A, B, 
<< 2. 

Discuss, as in the preceding exercise, the reductions which occur when 
a certain number of the sides or diagonals of the quadrilateral ABCD 
touch the fixed conic S. | 

27. The straight line whose equation in rectangular co-ordinates is 


xsina —ycosa=acoska, 


has as envelope, when a varies, a@ and & remaining constant, a hypo- 
cycloid or an epicycloid (§ 378). Find the locus of the points at which 
perpendicular tangents can be drawn to the envelope curve. Examine 
the particular cases k= 2, k =3. 


ips 


504 PLANE GEOMETRY. BOOK Iv. 


CHAPTER V* 
CONCERNING SIMILITUDE. 


389. We recall first the definition of two homothetic figures. 
Let A, B, C,--- (Fig. 256) be any system of points, situated 
in a plane; these points may be 
isolated or arranged on lines 
passing through O, taken arbi- 
trarily in the plane; draw to the 
various points of the system the 
half-lines OA, OB, OC, -+-, and 
take on these half-lines the points 
A', B', C'”, --» so that we have 

OA OBA200... 

OAL OB) 00. 
the system of points thus deter- 
mined is said to be similar to the proposed system and similarly 
placed. 

If the points A’, B’, C’, .-. were taken on the prolonga- 
tions of the half-lines in opposite directions, the two systems 
would be similar and oppositely placed. By a rotation about 
the point O through 180 de- 
grees, the second system will 
-\¢ coincide with one of the sys- 
tems similar and similarly 
placed with respect to A, B, 
C, «++ (Fig. 257). 

In order to abbreviate the 
expression, M. Chasles has 
called this similitude of form 
and of position homothetic, direct in the first case, and opposite 
in the second. The point O is called the center of similitude 





eee 9 





Fig. 257. 


CHAP. V. CONCERNING SIMILITUDE. 505 


or homothetic center of the two systems, the number & is the 
ratio of similitude, and the points A and A’ situated on the 
same half-line are called homologous points. If the ratio k 
vary from 0 to also the position of the center O of. simili- 
tude, one obtains all the systems homothetic to the given 
system. 

A system is similar to a given system, when it is equal to 
one of the systems homothetic to the given system. 


390. We know that all the curves homothetic to a given 
curve S with a single center of similitude O, taken at random 
in its plane, may be found. Consider next some examples. 

1° The curve S is a circle. If the center of the circle be 
taken as the center of similitude, the second curve will be a 
circle, whose radius can have any magnitude we wish. 

2° The curve S is a parabola (Fig. 258). The curve being 
referred to its axis and to the tangent at its 











vertex, the co-ordinates x and y of any point 7 
M of this curve satisfy the equation y? = 2 pa. 
If the vertex be taken as center of similitude = 
and «' and y' be the co-ordinates of the point 
M' homothetic to M, one has : i 
a ay 
et y! ae es.) ae 
/Fig. 258. 


whence # = kz', y = ky'; the homothetic curve 
2 ae 
will have the equation y? = ae it is a parabola whose pa- 


rameter can have as large a magnitude as is desired, on account 
of the arbitrary ratio k; it follows that any two parabolas are 
similar. 
2 2 
3° The curve S is the ellipse = cn 1. If the center of 
a 


the curve be taken as center of similitude, the homothetic 
curve, represented by the equation 


is an ellipse, whose axes can have any magnitude proportional 


506 PLANE GEOMETRY. BOOK IV. 


to those of the axes of the first ellipse. It follows that two 
ellipses are similar when their axes are proportional. 

The same theorem holds for the hyperbola. | 

4° The curve S is the logarithmic spiral p=ae™. If the 
pole be taken as the center of similitude, the homothetic curves 


have the equation p = = -- If one put k= e™, this equation 


becomes p= a™”-®; it represents the given spiral which has 
been revolved about the pole through the angle a It follows 
that the only curve similar to a logarithmic spiral is this spiral 
itself; to a point M of the curve corresponds another point M’ 
of the same curve, and this point ' can be taken at will, on 
account of the arbitrary number k. 


THE EQUATION OF HOMOTHETIC CURVES. 


391. Let 
(1) F@, y= 9 
be the equation of the curve S. Take the origin as center of 
. similitude and construct a curve S’ 
¥ - ; homothetic to the first with the ratio 
k. Let (a, y) be the co-ordinates of 
M any point M of the first curve, and 
eee | \ *' (a', y') those of the homologous point 
M' of the second curve, the similar 
triangles OPM, OP'M' give 
=the een % Be pe gle 
Fig. 259. ay’! «OM ? 
if «=a'k, y=y'k be substituted in equation (1), one has 
equation ; 


(2) Sf (kx', ky!) = 0, 


which represents all curves homothetic to the given curve and 
having the origin as homothetic center. Positive values of k 
in this equation will correspond to a direct homothetic trans- 
formation, and negative values to indirect. 

Allowing S to be fixed, transfer the curve S' in the plane, in 
such a manner that the origin O will fall in O' (p, q), and that 


~~ 








So 








i) 





‘ 


CHAP. V. CONCERNING SIMILITUDE. 507 


the axes will remain parallel to their primitive directions; the 
curve S' has as equation, referred to the axes O'X' and O'Y’, 


St (ke, ky') = 0, 
and, with respect to the fixed axes OX and OY, 


(3) Sl k(@ — p), k(y —q)]=0. 


In this new position, the curve S' is homothetic to the curve 8; 
because the radii vectores drawn from the points O and O' are 
parallel and in the constant ratio k. Equation (3) represents 
therefore all the curves homothetic to the given curve, whatever 
be the position of the center of similitude. 


392. At the same time that the origin is transferred to 0’, 
revolve the axes through the angle «; the curve S’ will then 
have an arbitrary position in the plane, and will be simply 
similar to the given curve. The curve S! referred to the vari- 
able axes O'X' and O'Y’ has the equation f(ka', ky')=0; by 
reason of the formulas of transformation, 


aw! = (a — p)cosa+(y—q) sina, 
y' = —(@ — p) sine +(y — 9) cos a, 


the axes being assumed rectangular, the equation of the curve 
with respect to the two fixed axes OX and OY is obtained. 
This equation represents all curves similar to the given curve. 


393. As an application, determine the conditions for which 
the two curves of the second degree, 


Ax’ + 2 Bry + Cy? +2 Dxe+2Ey+F=0, 
Ale? + 2 Blay + Cy? +2 D'x +2 E'y+ F'=0, 
are homothetic. The general equation of the curve homothetic 
to the first curve is (§ 391) 
Ak*x?’ + 2 Bk’axy + Ch°y? — 2 (Bk’q + Ak*p — Dk)ax 
— 2(Bk*p + Chéq — Ek)y + (Akep? +2 Blepq + Oke? 
— 2 Dkp — 2 Ekq + F)=0. 


508 PLANE GEOMETRY. — BOOK IV. 


In order that this equation be identical with the second, one 
must have 


— By Ap +2 — Bp— +4 





AR Oe: . 
ee ay Oe D' = E' 
Ap’ + 2 Bpqg+ CC? es ee Ee 





The elimination of the three quantities p, q, k from these five 
conditions will give two equations of condition; or the first 

iB? 
al Be 
ters to be eliminated, are simply the equations of condition 
required. Therefore, in order that two curves of the second 
degree be homothetic, it is necessary that the coefficients of the 
terms of the second degree be proportional. 


two equations — which do not involve the parame- 


394. It remains to inquire whether the parameters p, q, k 
are real or finite; we attack this question in the following 
manner: since the coefficients of the terms of the second de- 
gree are proportional, they can be made equal by multiplying 
all the terms of the second degree by a suitable factor; con- 
sider, therefore, the two equations under the form 


(4) Av? +2 Bey + Cy? +2 Dr+2 Hy+ F=90, 
(5) Al? +2 Blayt+ Oy+2D'e4+2 Ey+r=0 
Several cases may arise according to the sign of the quantity 
AC — B’. 
1° AC — B?=0. The two loci belong to the genus parab- 


ola; if these loci be parabolas, they are certainly similar, 
since all parabolas are similar; further, the axes of the two 


curves, having equal angular coefficients -% are parallel, 
and, consequently, the curyes are homothetic. 


2° AC —B?>0. The loci belong to the genus ellipse. If, 
in case of each curve, the axes be transferred parallel to them- 


CHAP. V. °* CONCERNING SIMILITUDE. 509 


selves to the center of the curve, equations (4) and (5) become 
(6) Aa? +2 Bay + Oy + H=0, | 
(7) Ax’? + 2 Bey + Cy’? + H'= 0. 

The axes of the curves, whose directions are determined by 


the equation tan 2@= 2 - (§ 139), are parallel; if the axes 





—C 
of co-ordinates be revolved through the angle a, equations (6) 
and (7) reduce to the form 

(8) Al’ + Cy + H =0, 

(9) Al’ + Cy + H'=0. 


The coefficients A' and C’, whose values are given by the equa- 
tions 





A+C'=A+C, A'—C'=4-ViB + (A— OF, 


have the same sign; in order that equations (8) and (9) repre- 
sent two real ellipses, it is necessary that the quantities H 
and H' have the same sign, and that, further, this sign be con- 
trary to that of A’ and C’. When this condition is fulfilled, 
the axes of the two ellipses having the same ratio rc the 
ellipses are homothetic. 

3° AC— B’<0. The two loci belong to the genus hyper- 
bola. On making the same transformation as in the preceding 
case, one is led to equations (8) and (9), in which 4! and C’ 
have contrary signs. When the quantities H and H' are dif- 
ferent from zero, each of the equations represents a hyperbola. 
If H and H' have the same sign, the real axes of the two 
curves are parallel; and, since the ratio of the axes is the same, 
the curves are homothetic. When H and H' have contrary 
signs, one of the hyperbolas is similar to the conjugate of the 
other. In the two cases the curves have parallel asymptotes. 

It follows from what precedes that, when two equations of 
the second degree have coefficients of the terms of the second degree 
proportional, and represent real curves, these curves are homo- 
thetic; excepting, when the curves are hyperbolas, it can happen 
that one is homothetic to the conjugate of the other. 


510 PLANE GEOMETRY. BOOK IV. 


GENERAL EQUATION OF A SPECIES OF CURVES. 


395. Curves of the same species are curves comprised within 
the same geometric definition, and which do not differ from one 
another excepting in the values assigned to the parameters which 
are involved in the general definition. The general equation 
of the curves of the species considered is an equation which, in 
a system of co-ordinates, gives all these curves, whatever be 
their position in the plane, when various values are assigned 
to the variable parameters which it involves. Thus, when the 
fixed axes are rectangular, the general equation of the species 
circle is (w—a)’?+(y—b)?=7*, This equation involves three 
variable parameters, namely, the radius r and the two co- 
ordinates of the center, a and b. 

Accordingly, one seeks the equation of the curve with re- 
spect to particular axes, which are chosen to simplify the 
calculation; then, to obtain the general equation, one refers it 
to the fixed axes by a transformation of co-ordinates. 

The lemniscate has been defined as the locus of points such 
that the product of the distances of each of them from two 
points # and F" is equal to 
the square of half of the 
distance FF" (Fig. 260). If 
the mid-point O' of the 
straight line FF" be taken 
as origin, O'F and a_per- 
pendicular to O'F as axes 
of co-ordinates, and if the 

Fig. 260. | distance F'F' be designated 
by 2c, the curve, referred 
to these particular variable axes, is represented by the 
equation (§ 339) 
(a? “a y')? oe, cy? an ar!) = 0. 

The curve may be referred finally to the fixed rectangular 

axes OX, OY, by means of the formulas of transformation 


¥ 











x'=(x —a)cosa+(y—b)sina, 


y'=—(«#—a)sina+(y — b)cosa, 


CHAP. V. CONCERNING SIMILITUDE. 511 


in which @ and 6 designate the co-ordinates of the point O! 
with respect to the fixed axes, and « the angle formed by O'F 
with OX. One is thus led to an equation 


F(a, y, ¢, a, b, «)=0 


involving four arbitrary parameters, and which represents all 
lemniscates; it is the general equation of the species. 

The general equation of a species of curves, with respect to 
fixed axes, contains three parameters more than the equation of 
the same curves referred to the axes associated with the curves 
‘in a determined manner. Let n be the total number of param- 
eters, this system of parameters could always be replaced by 
another system, such that the variation of three of them causes 
only the curve to be displaced in its plane, while the variation 
of the n — 3 others gives the different curves. 


396. The number of points, and, in general, the number of 
conditions necessary to completely determine a curve of given 
species, is equal to the number of arbitrary parameters which 
the general equation of the species involves. The remarks 
made concerning the curves of the second degree (§ 283), on 
multiple conditions, are applicable here. It is often important 
to previously show that the parameters which enter in the 
equation cannot be reduced to a smaller number. 

Consider, for example, the locus of points such that the 
ratio of their distances from two fixed points whose co-ordinates 
are (a, b), (a', b') is constant and equal to k. This locus, whose 
equation is 


GQ) @—a)’+y—b)—B[@—a'l? + y—b?]=0, 


is a circle. Equation (1) involves five parameters; but by 
developing and by arranging the terms, it becomes 











a —k?a' b—k’b' | a +b°—k(a"? +b") 
ee ee re a 


Three of the coefficients only involve the parameters; if they 
be replaced by A, B, C, equation (2) takes the form 


(3) e+y+ Ax+ By+C=0. 


(2) w+y?—2 


512 PLANE GEOMETRY. BOOK IV. 


Three points are sufficient to determine the coefficients A, B, 
C, and, consequently, the circumference. If one wish then to 
obtain a, b, a’, b', k, one will have a system of three equations 
in five unknown quantities; the solution will be indeterminate, 
and two of the unknown quantities could be chosen at will; 
this signifies that, for the same circumference, one can find an 
infinity of pairs of two points such that the ratio of their dis- 
tances from each of the points of the circumference from these 
two fixed points is constant. 


397. The geometric definition of a species of curves em-— 
braces within it the number of arbitrary parameters involved 
in its general equation. The definition of the circle assumes 
that we know its center, whose position is determined by its 
two co-ordinates, and by the length of the radius, in all three 
constants or arbitrary parameters. The definition of the lem- 
niscate assumes that two fixed points are given, which are 
equivalent to four fixed constants. In case of the ellipse, 
since it is necessary to know the sum of the radii vectores, 
the curve involves five parameters; in the definition of the 
spiral of Archimedes one is given a pole, which is equivalent 
to two constants, the position of the straight line at the instant 
when the movable point passes through the pole, and a ratio: 
in all four constants. 


398. An integral polynomial of the degree m jm the two 


variables 2 and y contains Mus a Bie terms; it follows 





points are necessary to 





that sae lad or ee) 
2 2 


define an algebraic curve of the mth degree. For example, 
nine points are necessary to determine a curve of the third 
degree, fourteen points to define a curve of the fourth degree. 

Consequently, that every equation of the third ae can 
be written in the form 


(4) Hy Avs a kB’y = 0 


where @;, 2, &, 8, y are linear functions of the first degree, and 
k is an arbitrary parameter because this equation contains 


CHAP. V. CONCERNING SIMILITUDE. 513° 


eleven arbitrary parameters; one can choose at will one of the 
five linear functions, since there will still remain nine param- 
eters. The three straight lines «,=0, a,=0, a, =0 are tan- 
gents to the curve at the points where they are intersected by 
the straight line B=0; and the points where these tangents 
are intersected by the straight line y =0 belong moreover to 
the curve. On taking £B at will, the following theorem is 
deduced: if a curve of the third degree be intersected by any 
straight line B = 0, and tangents be drawn to the curve at the 
three points of intersection, each of the tangents intersects the 
curve in one additional point, and these three points he on a 
straight line. On taking y at will, we get another theorem: 
if a curve of the third degree be intersected by a straight line 
y=0, and if through each of the points of intersection tan- 
gents be drawn to the curve, the points of contact of these 
tangents lie on a straight line. 

Suppose that the straight line 8B = 0 be removed to infinity, 
the three tangents a, = 0, a, =0, a, = 0 have as limits the three 
asymptotes of the curve; each of these asymptotes intersects 
the curve in one point only, and these three points lie on a 
straight line. 

The equation 
(5) , (109%, — kB? = 0, 


which involves nine arbitrary parameters, represents all the 
curves of the third degree. The three points where the 
straight line 8 = 0 intersects the curve are points of inflection ; 
the tangents at these points are the straight lines 


aut, a=), Got. 


All the curves of the third degree can moreover be repre- 
sented by the equation 


(6) a (a — ay)(a — by) — kp’y = 0, 

which involves nine arbitrary parameters. The straight line 
y=Ois the tangent at the point of inflection (« = 0, y=90); 
the three tangents «= 0, «—ay=0, a—by=0 at the points 
where the straight line @=0 intersects the curve pass 


through this point of inflection. On taking the perspective on 
2K 


514 PLANE GEOMETRY. BOOK IV. 


a plane, any straight line we wish may be removed to infinity; 
for example, the tangent y = 0 at the point of inflection; it is 
sufficient to make y = 1 in the equation which reduces to the 
form a(a— a) (a—b)—kf#’?=0; if the straight line B = 0 be 
taken as the x-axis and the straight line a=0 as the y-axis, one 
obtains the equation ky’? =x(x—a)(x—b), which has been 
discussed in § 337. 


399. Every equation of the fourth degree can be written in 
the form 
(7) 01 0x30, — kB’ = 0, 


when ¢ is a polynomial of the second degree; because this 
equation contains sixteen parameters, one can take £B at will, 
since there will remain fourteen parameters. Thus when a 
curve of the fourth degree is intersected by any straight line 
8 =0, and tangents be drawn to the curve at the four points of 
intersection, if the two other points in which each tangent in- 
tersects the curve be taken, there will be eight points of the 
curve situated on the conic ¢ = 0. 

A curve of the fourth degree has four asymptotes; each of 
them intersects the curve in two points; the eight points of 
intersection lie on a conic. 


CONDITION OF SIMILITUDE OF Two FIGURES. 


400. Consider a series of curves of the same species, in 
whose definition there enters only the linear parameter A, 
whose measure is represented by a, and let 


(1) S (&, Y; a) = 0 


be the equation which represents all these curves, without 
reference to their position in the plane. If equation (1) were 
obtained without specifying the linear unit, it is necessarily 
homogeneous with respect to a, y, a. When the linear param- 
eter A changes, which happens, the unit remaining the same, 
if the number a vary, equation (1) defines a series of homo- 


CHAP. V. CONCERNING SIMILITUDE. 515 


thetic curves. In fact, let A) be a parameter whose measure is 
a; to this parameter corresponds the particular curve 


(2) Sf (@, Y; Alo) = 0. 


The curves homothetic to curve (2), the origin being the center 
of homothétie and k an arbitrary ratio, are represented by the 
equation (§ 391) - 


(3) S (kx, ky, a) = 0. 
Let A, be a second parameter which is measured by a such 
that = k, equation (3) may be written 
Sf (ka, ky, kay) = k"f (a, y, a) = 0, 
(4) or IE, ¥; %) = 9, 


which shows that the curves homothetic to curve (2) are the 
various curves which are obtained by varying the parameter A. 


401. In general, suppose that n linear parameters A, B, --- 
fail to define all the curves of the same species, without regard 
to their position in the plane; let a, b, c,--- be the measures of 
these parameters with respect to an arbitrary unit; the equa- 
tion of the curves of the species 


(5) F(X, y, a, b, +++) =0 

will be homogeneous with respect to a, y, a, b,---. The curves 
defined by equation (5) and which correspond to the two series 
of proportional parameters Ay, By, +.» and Aj, B,, ++» are homo- 
thetic ; because, if k represent the ratio of the parameters 


Ay dy 
a 6 


the curves homothetic to the curve 
I (&; Y, Ao; bo, see) = 0 
are represented by the equation 
S (kx, ky, ka, kb, +++) 2 k*f (a, Y, Ay, bi, vee) ans 0, 
or I (&; Y, A, bd, vee) = Q), 


ee =i, 


516 PLANE GEOMETRY. BOOK IV. 


It follows from what precedes that when the curve, without 
regard to its position in the plane, is defined by a single mag- 
nitude, that all the curves of the species are similar. Thus, 
the circle being defined by its radius, the parabola by the 
distance of the focus from the directrix, the lemniscate by 
the distance between the foci, the spiral of Archimedes by the 
length intercepted on the yadius vector between two succes- 
sive helices, all circumferences are similar, and similarly all 
parabolas, all lemniscates, ete. 

The ellipse being defined by its two axes, the condition of 
similarity of two ellipses is that these axes are proportional, 
as we have already learned in § 390. The same is true of two 
hyperbolas. 


CHAP. VI. GRAPHIC SOLUTION OF EQUATIONS. 517 


CHAPTER YVI* 
GRAPHIC SOLUTION OF EQUATIONS. 


402. Consider two equations in two unknown quantities 
x and y 


(1) p(x, y) = 9, (2) Wy (a, y) = 90; 


each of them defines a curve. For this system of two equa- 
tions can be substituted an infinity of equivalent systems of 
equations; consider in particular a system 


(3) x (2, y) = 9, (4) St (a) = 0, 


one of which only contains the variable y, a system which may 
be obtained by eliminating y from the two given equations. 
The real roots of equation (4) are the abscissas of the points 
common to the two curves (1) and (2).. And yet, if the system 
of equations (3) and (4) were satisfied by a pair of values of 
the form =a, y=B+ yi, in which a, B, y are real, these 
values would satisfy the system of equations (1) and (2); but 
the quantity « would not be the abscissa of a real point com- 
mon to the two curves. The exception which we have pointed 
out never occurs when the equation y(a, y) =0 is an algebraic 
equation which involves only y to the first degree. 

When one wishes to solve an equation f(x)=0 in a single 
unknown quantity, the curves determined by (1) and (2) may 
be selected in an infinity of different ways. The only con- 
dition to be fulfilled is that the elimination of y between equa- 
tions (1) and (2) give the proposed equation. A first combina- 
tion is y= f(“), y= 0, which leads one to consider the values 
of the unknown quantity as the abscissas of the points of inter- 
section of the curve y= f(x) with the z-axis. This combina- 
tion is rarely the most simple. It is proven in algebra that 


518 PLANE GEOMETRY. BOOK Iv. 


if an unknown quantity y be eliminated from two algebraic 
equations in two unknown quantities whose degrees are m 
and n, the resulting equation in « is at most of the degree mn. 
Consequently, if the proposed equation be algebraic and one 
wish to obtain its roots by the intersection of algebraic curves, 
the product of the degree of the equation of the two curves 
would be equal to the degree of the equation to be solved. 
We shall apply this method to the solution of the Seauaton of 
the fourth degree. 


403. The equation of the fourth degree may be easily re- 
duced to the form 


(5) e+ par+qurutir=0; 


it may be regarded as the result of the elimination of y be- 
tween the two equations of the second degree 


(6) x? —my=0, (7) my? + pmy + gqx+r=0, 


each of which defines a parabola. Since equation (6) involves 
y only to the first degree, all the real roots of equation (5) are 
the abscissas of real points common to the two curves. 

One can substitute for the parabola (7) another curve of the 
second degree which passes through the intersection of the 
curves (6) and (7). The general equation of the second degree 
satisfying this condition (§ 277) is 


ft 
c 


ae ke? + my? + qu +m(p—k)y+r=0, 


k being an arbitrary parameter. If one put k = m?, the curve 
(8) would be simply a circle; the co-ordinates a and b of the 
center and the radius # of this circle are given by the formulas 


When the value of 2? is positive, equation (8) ee. a real 
circle, and the real roots of equation (5) are the abscissas of the 
points of intersection of this circle and parabola (6). When 
the value of f? is negative, equation (8) cannot have real 
solutions (§ 85); the same is true of the system of equations 


CHAP. VI. GRAPHIC SOLUTION OF EQUATIONS. 519 


(6) and (7), or of the equivalent system of equations (9) and 
(6); the four roots of equation (5) will be imaginary. 


404. Consider next the equation of the third degree reduced 
to the form 
e+ petq=0. 


If this equation be multiplied by 2, which introduces the root 
x =, an equation of the fourth degree is obtained, 


a* + pa’ + qu = 0, 


to which the preceding method is applicable. The value of R?, 
being in this case equal to a?+ 0’, is always positive. The 
circle and the parabola pass through the origin of co-ordinates ; 
the abscissa of this point is the root «= 0, which one should 
remove. 

The same parabola a? —my=0 may serve for the solution 
of all equations of the third or of the fourth degree; the circle 
only changes depending upon the value of the coefficients of 
the proposed equations. This method can be employed with 
advantage when one would solve successively a great number 
of equations; then a parabola having an arbitrary parameter 
is traced with great care; and, in each particular example, it 
only remains to determine the circle. 


_ 405. When the unknown quantity x is a line, and the unit 
of length has not been specified, the equation f(7) =0 is a 
homogeneous equation in the unknown quantity x and the 
various known lines. In case the equation is of the fourth 
degree, if the coefficients p, g, 7 be rational functions, or irra- 
tional functions of the second degree of given lengths, on 
taking an arbitrary length for the parameter m of the parabola, 
the co-ordinates of the center and the radius of the circle could 
be constructed with the rule and the compass. 

But if the equation be a numerical equation, that is if the 
coefficients be given numbers, a definite value is given to m; 
for example, one would put m = 1, and construct the parabola 
and the circle by means of an arbitrary scale; the abscissa of 


520 | PLANE GEOMETRY. BOOK IV. 


one of the points of intersection measured by the same scale, 
will give the value of the unknown quantity a. 

We know that the solution of two equations of the second 
degree in two unknown quantities 2 and y, or the determina- 
tion of the points of intersection of two curves of the second 
degree, reduces to the solution of an equation of the fourth 
degree in one unknown quantity. This solution could there- 
fore be accomplished by means of a definite parabola and a 
circle. Accordingly, if one of the curves of the second degree 
be already traced, it can be used with the circle. 


406. Exampere I.—Draw through a given point P whose co-ordinates 
are x; and y; a normal to a parabola y? —2px=0. The co-ordinates x 
and y of the foot of the normal are determined by the system of equations 


y—2pr=0, xy—(%1—p)y — py =0. 
If all the terms of the last equation be multiplied by y, and if 2px be 


substituted for y?, a new parabola x? — (x#1—p)x— “AY _ 0 is obtained ; on 


2 





adding the equations of the ‘two parabolas member to member, one 


obtains the circle 2? + y? —(a, + p)x — oa =0. The points where this 
circle intersects the given parabola are the feet of the normals (§ 306). 





407. Exampre II.— Solve the numerical equation #3 ~ x2 —7 =0. 

: Construct by means of an accurately 

made scale, the parabola 22 = y; 

| describe a circle whose center C has 

=| PRT the co-ordinates a= 4, b=1, and 

ue : SN which passes through the origin ; 

/ \ this circle intersects the parabola in 

v ‘\ one additional point M; therefore 

\ the proposed equation has but one 

real root, the abscissa OP of the 

point M (Fig. 261). By measuring 

\ P a x this length by means of the scale 
/ here employéd, we find x = 2.09. 


re 








8 
b 





ee _o? ExampLeE III. —Solve the equa- 

ieee tion #3 —52+1=0. Construct 
the circle whose center C has the 
co-ordinates a =— 4, b= 8 and which passes through the origin: this 
circle intersects the parabola in three points; it follows that the equation 


Fig. 261. 


CHAP. VI. GRAPHIC SOLUTION OF EQUATIONS. 521 


has three real roots; on measuring the abscissas, one finds that the two 
positive roots are 0.20 and 2.18. 


408. Exampie IV. — Consider the transcendental equation 
xtangx=1. 


This equation is the result of the elimination of y between the two equa- 
tions 
y=tangz, sy=1. 


The first represents a curve composed of an infinitude of equal branches 
which have asymptotes perpendicular to the x-axis ; the second an equi- 
lateral hyperbola (Fig. 262). It is evident that the right-hand branch of 


Di 























Fig. 262, 


the hyperbola intersects, at least once, each of the branches OA, B’ J’, ... 
of the transcendental curve; moreover there is but a single point of 
intersection on each branch, because, when «x varies, the ordinates of the 
two curves vary in contrary directions; if these ordinates be equal for a 
certain value of x, they are necessarily unequal for every other value. 
The roots of the equation are equal in pairs with contrary signs ; there 


is in the first place a root situated between 0 and = a second root between 
m7 and “2, a third between 27 and om etc., ...; the number of roots is 


infinite. On calling x, the nth root, the difference between «x and 
(n —1)m is very small when n is very large. The curve gives, for the 
value of the first root, 0.86. 


522 PLANE GEOMETRY. BOOK Iv. 


The equations y = tang : ~ n), y = x could also be discussed in this 
manner on putting 5 —x=2', y=tangz’, y= - —zx'; the hyperbola 


would be replaced by a straight line. 


409. Remarxs.—The graphic methods which we have 
described do not give the values of the unknown quantities 
with any great precision; one should not expect an approxi- 
mation nearer than the hundredth part of the root. 

One often attacks the problem by magnifying the traces of 
the two curves in order to determine the number of real roots 
of an equation. But, so long as the form of the two curves 
is not studied with care, no rigorous conclusions can be de- 
duced from this discussion. In general, the discussion of the 
curves and the determination of the points of intersection 
offer the same difficulties as the problem proposed. 


CHAP. VII. UNICURSAL CURVES. 523 


CHAPTER VII* 
NOTIONS CONCERNING UNICURSAL CURVES. 


410. We have learned (page 339) what facilities offer them- 
selves for studying a curve when the co-ordinates of one of its 
points can be conveniently expressed as a function of a param- 
eter. From this point of view, the most simple algebraic 
curves are those whose co-ordinates can be expressed as a 
rational function of a parameter. These particular curves 
have been called unicursal curves. 

The theory of unicursal curves was formed by Chasles and 
Clebsch. The results which we give have been taken in most 
part from the Memoirs of Clebsch (Crelle Journal, v. LXIV. 
and LXXIII.) and from the lectures on geometry by Clebsch, 
published by Lindemann. 

I. A curve of the order m with a multiple point of the order 
m — 1 is unicursal. 

If the multiple point be chosen as origin, the equation of 
the curve becomes (§ 354) 


m1 (X,Y) + bn (@ y) = 9, 
where ¢,,_,; and ¢,, designate homogeneous polynomials in 
x and y of the degrees m andm—rz. A variable straight line 
y = ta, 


passing through the origin, intersects the curve in m — 1 points 
coincident with the origin, and in one other variable point M 
whose co-ordinates are given by the equations 


Pin 1 (1, t) thin (1, t) 
7 ES —_— — ’ 
©) i 7 8) 
that is, are rational functions of the parameter ¢, of the degree 
m, with the same denominator. To each value of ¢ corresponds 





524 PLANE GEOMETRY. BOOK IV. 


on the curve a single point, and conversely to each point M of 
the curve distinct from the multiple point x = 0, y = 0 there cor- 


responds a single value of ¢t, t = z, equal to the angular coeffi- 


cient of the straight line OM; as for the multiple point «= 0, 
y = 0, it is obtained by m — 1 values of t, roots of the equation 


Pin) (A, t) a 0. 


It can still be said that, if « and y be the co-ordinates of a 
point of the curve distinct from the multiple point, equation 


(1) of the degree m in ¢ has a single common root t= y and 
oG 


if «=0, y = 0, these equations aa m — I given roots common 
with the equation ¢,,_; (1, t) = 

The following curves oie in this aatecone of unicursal 
curves: the conics (m = 2); the curves of the third order with 
a cusp or with a double point (m = 3), for example, the cissoid 
and strophoid; then the curves of the fourth order with a 
triple point, for example, the curve constructed on page 
422... ete.-»-. We shall learn later that there exist other 
unicursal curves of the fourth order, namely, curves having 
three singular points, double points, or cusps. 


II. UNIcURSAL CURVES OF THE THIRD ORDER. 
Let 
(1) Aa’? + Buy + Cy? + Da? + Ex’y + Foy’ + Gy =90 


be the equation of a curve of the third order having a singular 
point at the origin. If one make y=¢ta, one obtains, for the 
co-ordinates of a point M of the curves, the following expres- 








sions : ' 
(2) i eo ee a Bt 

ere One ~ © 
where ¢(t) = D+ Et+ F’ + Ge. 


The values of ¢, corresponding to the singular point situated at 
the origin, are roots of the equation of the second degree 


A+ B+ CP =0 


CHAP. VII. UNICURSAL CURVES. 525 


which give the angular coefficients of the tangents at the 
origin; the singular point will be a double point or a cusp 
according as these roots are unequal or equal (§ 354). 


A CURVE OF THE THIRD ORDER WITH ONE CUSP. 


Suppose first that these roots are equal: let a be their 
common value; one will have 


A+ Bt+ C?'= Ct — a)’, 

whence 
: ete 2 aa 2 
(3) el Cti—a)y  Ctt—a) 


See SO 


Take on the curve three points corresponding to the values 
ty, ta, ts of the parameter, and determine the necessary and 
sufficient condition in order that these points be on a straight 
line. 

Let ux +vy+w=0 


be the equation of a straight line which does not pass through 
the cusp. The values 4, t., ¢; of the parameter ¢ corresponding 
to the points of intersection of this straight line with the curve 
are the roots of the equation of the third degree found by 
substituting in the equation of the straight line the values of 
« and y in expression (3). By this substitution, the trino- 
mial ux + vy + w becomes a rational function in ¢ of which the 
denominator is #(¢) and of which the numerator becomes zero 
for the values ¢,, t., t, of t. One has therefore by this substitu- 
tion, identically 


Kit —t) ¢—4)G—&) 
$(¢) 
K being a constant depending on wu, v, w. On taking the deriva- 


tives of the two members of this identity with respect to t, we 
will have another identity 


oo keke Ga) 1 | FE 1 8 
UL, + VY, = b(t) laaticetice ae | 


Ux + vy +w= 








526 _ PLANE GEOMETRY. BOOK IV. 


Substitute, in this last identity in ¢, the value t=a fort, 
which corresponds to the cusp. Since, for this value, 2’, and 
y', become zero and the factors 


are not zero, the secant not passing through the cusp, it follows 
that 
1 1 1 '(a) 
4 _ 
@) Py Ser Sy a—t, (a) 








Since this relation is independent of wu, v, w it holds neces- 
sarily between the parameters t,, t,t; of the three points on 
the straight line. Conversely, if this relation be satisfied, the 
three corresponding points are on a straight line, for on eall- 
ing t'; the parameter of a point which is on this straight line 
passing through the first two, one has 

1 1 1 g(a) 


a—t,'a—t a—t's  (a)’ 








whence by comparison with (4), t'; = ty. 

Point of Inflection. The tangent at a point of inflection 
intersects the curve in three points coincident with the point 
of contact: if therefore one call 6 the parameter of a point of 
inflection, one will have, by putting 





t, = te = t. — 6 
in formula (4) oe 
3 _4'@) 

a—-O (a) 


The curve has, therefore, a single point of inflection. 
Points on a conic. A conic whose equation is 


I (@, y= aa? + Bay + yy? + du + ey+y=0, 


and which does not pass through the cusp, intersects the curve 
in six points, whose parameters ¢,, ta, ts, ty, ts, tg are roots of the 
equation of the sixth degree obtained by replacing, in the 
equation of the conic, x and y by their values (3). By this 
substitution f(2, y) becomes a rational fraction of the sixth 
degree in ¢t, whose denominator is ¢7(¢) and whose numerator 


CHAP. VII. UNICURSAL CURVES. 527 


is a polynomial which becomes zero for the values ¢), to, tzy tyy t5y fg 
of t. Hence follows the identity 


H(t — t,)(t —t)---(t— t) 
$*(t) 
where H is a certain constant independent of t; then, on tak- 


ing the derivatives of both members with respect to ¢, one has 
the identity 


St (a, Y) aig 











H(t.— t,) (¢ — t,)---(¢—t,)[ 1 1 
' Jena 
ISH: + fy’ = $’(t) t—t, ne i 
1 290), 
Psa ey | 
and on putting in this last identity t = a, one has 
1 1 2: 1 1 1 2 ¢' 
(5) ee ee 











a—t a—t, G—t, a—t, a—t;'a—t o(a)’ 


the necessary condition in order that the six points correspond- 
ing to these six values of ¢ be on a conic. It may be seen, as 
above, that this condition is sufficient. 

These considerations are, in particular, applicable to the 
pedal or to the inverse curve of a parabola, with respect to a 
_ point of the parabola. 


CURVES OF THE THIRD ORDER WITH A DOUBLE POINT. 
Let a and } be the roots of the equation 
A+ Bt+ Cf =0, 


which gives the angular coefficients of the tangents at the 
singular point (page 435); if these roots be imaginary, the 
double point is isolated. The expressions for the co-ordinates 
wand y of a point of the curve are 

(6) en C(t — a) (t — b) _ —Ct(t—a)(t— b) 


(t) i $(¢) 


Find the necessary and sufficient condition, in order that the 
three points corresponding to the values t,, t, t; of the parame- 








528 PLANE GEOMETRY. BOOK IV. 


ter lie on a straight line. Let ua + vy + w be the first member 
of an equation of a straight line which does not pass through 
a double point; if 2 and y be replaced by their expressions in 
t, this first member becomes a rational fraction, whose denomi- 
nator is #(t), and whose numerator becomes zero for the values 
t,, to, t; of t. One has therefore by this substitution the identi- 
cal equation : 
K(t — t)(t — ts) (t — t) 


ux+ vy +w= $0) 





Substitute successively for t, in this identity, the values a 
and b corresponding to a double point; since x and y become 
zero for either of these two values, it follows 


ae K (a — t;)(a — t,)(a@ — ts) 
a $ (a) 


K(b—t)(b — t)(0 — &) 
o (d) 





2 





whence by division 


(7) (a — t,)(a — ty) (a — bs) 2 (a) 
(—4)(b —t)(0—%) $0) 


This is the necessary condition, that three points in a straight 
line lie on the curve; it may be shown, as on page 507, that it 
is sufficient. os 

If the double point be real (the pedal of a parabola with 
respect to an external point, the inverse curve of a hyperbola 
with respect to a point of a curve), the constants a, b, (a), > (d) 
which enter in relation (7) are real. If the double point is 
isolated (the pedal of a parabola with respect to an interior 
point, the inverse curve of an ellipse with respect to a point of 
the curve) these constants a and b on the one hand, ¢(@) and 
¢ (b) on the other, are conjugate imaginaries. Relation (7) can 
then be replaced by another which involves only real elements. 
For this purpose suppose 





a=ptiq, b=p— gq, pep = cosh + isin2d 


CHAP. VIL. UNICURSAL CURVES. 529 


when 2) designates the argument of the constant quantity 


se whose modulus is 1, since ¢(a) and (5) are conjugate 


imaginaries. Put 
a—t cosr+isinr 


oe —— = cos2r+isin2r; 
b—t cosr—itsint 


 sesgs cot 7, 





to each value of r corresponds one value of ¢; to each value of 
t correspond an infinitude of values of 7+ differing from each 
other by multiples of z. Call 7, t., 73 the three values of r 
which correspond to the parameters ¢,, t,, ts of the three points 
on the straight line; relation (7) gives us 


cos 2 (7, + t2 +73) + isin 2 (7, + 72+ 73)= cos 2A +47sin2A, 
whence | 
(8) | +17 +7,=A+hkz, 


where kis any integer. On making in these formulas (7) and 
(8) 4,=4,=1,=1, or 7, = 7;= 73 = 7, one finds the value of the 
parameters ¢ or r which correspond to points of inflection. 
One can easily demonstrate that: there are three points of 
inflection ; these points lie on a straight line. If a and b be real, 
one only of these points is real; if a and b be imaginary, the three 
points of inflection are real. 

The necessary and sufficient condition in order that six points 
of the curve lie on a conic can be written 


a an Cee EEN CEE CEES 
(b—h)(b —4)(6 4) —4)G—t)G—h) Lb) 
or if a and b be imaginary, 

Trt t+ ts t+ +75 + 175= 2A + kz. 


III. The curves which we have thus far studied constitute 
all unicursal curves of the third order. In fact, let 


BE) 2) 
(9) oP Riry I= RO 





be the expressions of the co-ordinates of a point of the curve, 
where P(t), Q(), R() designate polynomials of the third 
ne 2% 


5380 PLANE GEOMETRY. BOOK IV. 


degree in ¢ which do not have a common divisor. To each 
value of ¢ corresponds a single point of the curve; suppose that 
conversely, save for certain special points finite in number, to 
each point (a, y) of this curve corresponds one value only of ¢. 
This is equivalent to saying that the equations 


(10) P)_ Pt) Q0_ 
BORG) RO Fe) 





do not have more than a finite number of solutions in which ¢ 
is different from t', or, moreover, that there is but a finite num- 
ber of values of w and y for which the equations in ¢ 


(9) zR()— P()=0, y¥RO-QH=0 


have two common roots. Then the curve defined by equations 
(9) is of the third order and has one singular point; equations 
(9) have two common roots when the point (2, y) coincides with 
this singular point. 

Then the curve is of the third order, for on changing the 
values of the parameter corresponding to the points of inter- 
section of the curve with a straight line, one gets an equation 
of the third degree. Whence, the equation of the tangent at 
the point with the parameter ¢ is ($ 341) 


y| 
er ae Gimme. 


that is, after introducing the expressions of x and y in f, 
X(RQ'— QR')+ Y(PR'— RP')+ QP'— PQ' = 0. 


In the combination such as RQ' — QR’, the term in ? disap- 
pears, and the equation of the tangent contains ¢ to the fourth 
power at most. If, therefore, one seek the values of ¢ corre- 
sponding to the points of contact of the tangents drawn from 
a point to the curve, one finds at most four values for t. The 
curve considered is therefore, at most, of the fourth class; 
it has a singular point, because a curve of the third order 
without a singular point is of the sixth class. 

IV. The following proposition concerning curves of the 
fourth order will be proven: | 


CHAP. VII. UNICURSAL CURVES. 531 


The curve of the fourth order with three singular points 
(node or cusps) is unicursal. Let A, B, C be the singular 
points, and D a fixed point taken on the curves. A variable 
conic 

S+tS,=0, 


which passes through the four points A, B, C, D, intersects the 
curve in eight points, seven of which are fixed; namely, two 
coincident with A, two with B, two with C, and one with D. 
The equation of the eighth degree, which gives the abscissas 
of the points of intersection of the conic and of the curve, 
has therefore seven fixed roots independent of t; one could 
suppress these roots by division, and there would remain an 
equation of the first degree in @ expressing x as a rational 
Junction of t. Similarly, y may be obtained as a rational func- 
tion of t. To each point of the curve different from the points 
A, B, C corresponds a value of the parameter ¢ given by the 


equation 


to the double points correspond two values of the parameter, 
which become’equal when the double point becomes a cusp. 

In this category of curves belong the lemniscate, the hypo- 
cycloid with three cusps, the pedals or the inverse curves of 
ag ellipse, and of a hyperbola with respect to a point not 
situated on these curves. 

It can be demonstrated that, conversely, every unicursal 
curve of the fourth order has a triple point or three singular 
points (node or cusp). 

V. Since a curve of the third order cannot have two and a 
curve of the fourth order cannot have four singular points, 
the unicursal curves of these orders are those which have the 
maximum number of singular points. This theorem can be 
made general for unicursal curves of all degrees. In fact, it 
can be proven that: 

A unicursal curve of the nth order, which does not have 
higher. singularities than nodes and cusps, may have nodes 


(n — 1)(n — 2) 
2 





or cusps in number; and, conversely, a curve 


532 PLANE GEOMETRY. BOOK IV. 





of the nth order with re ee nodes or cusps is uni- 


cursal. 





(n — 1)(n — 2) 
Z 


singular points which a curve of the nth order can have with- 
out decomposition. 

If one be given the expressions of the co-ordinates of a point 
of an unicursal curve as a rational function of a parameter 


(11) c= (t), oe y(t), 
the values of t corresponding to the singular points are found 
by seeking the solutions common to the two equations 


$OH= $C), YO=VO), 


in which ¢ is different from t'. 

It is to be noticed that the equation of the curve is obtained by 
eliminating ¢ between equations (11); that is, by expressing the 
conditions that these equations have a common root int. If 
the point (a, y) be an ordinary point of the curve, equations 
(11) do not have more than one common root; if the point 
(x, y) coincide with a singular point, they have two or more 
roots in common, which are the values of ¢ corresponding to 
the singular point. The singular points are therefore found 
by seeking the positions of the point (a, y) for which equa- 
tions (11) in ¢ have two or more roots in common. , 


This number, , 1s the maximum number of 


EXERCISES. 


1. A point M is taken on a curve of the third order with a cusp. The 
tangent at M intersects the curve in a point Mj, the tangent in M, inter- 
sects it in Mo, the tangent at J, intersects it in Ms, etc., ++. Show that 
the nth point M, thus determined, when n is indefinitely increased, 
approaches the cusp. 

From the point Ma tangent can be drawn to the curve. Let M’ be its 
point of contact ; from the point M! a second tangent can be drawn to it ; 
let Mz be its point of contact, etc.,---. Show that the nth point M™ thus 
determined approaches the point of inflection. 


2. Let M be a point on a curve of the third order with a double point. 
The tangent at M intersects the curve at M,, the tangent at M, intersects 
it at My, etc.,---; let M, be the nth point thus determined. 


CHAP. VII. UNICURSAL CURVES. 538 


If the tangents at the double point be real, the point M, approaches, 
when n is increased without limit, the double point. If these tangents 
be imaginary, it can happen that the point M, coincides with the point of 
departure M. One will then have a polygon of n sides whose vertices 
lie on the curve and the sides tangent to the curve. What positions 
is it necessary that the point M should take in order that this should 
happen? Study the particular cases n=8, 4, 5. (Durége, Math. 
Annalen, Erster Band.) 


3. From a point of inflection J, of a curve of the third order, one can 
draw, at the double point, a tangent 77 to this curve having its point of 
contact at T. Show that the straight lines which join the two points I 
and 7 to the double point are harmonic conjugates with respect to the 
tangents at the double point. 


2 2 
4. The co-ordinates of a point of the ellipse nee 1=0 can be 
a 


2 
expressed as a function of an angle ¢ by the formulas «= acos¢, 
y=bsing. Show that the necessary and sufficient condition, in order 
that four points of the ellipse correspond to the values 41, $2, $3, ¢4 of 
¢ should lie on the same circle, is 


o1 + 2+ $3 + $4 = 2k. 


Through a point Jf taken on the ellipse three circles osculating the 
ellipse can be passed (without intersecting the one which has its point of 
contact at M); prove that the points of contact of these three circles lie 
on a circle which passes through 

At the point M draw a circle osculating the ellipse; let I, be the 
point in which this circle intersects the curve; the osculating circle at 
M, intersects the ellipse at M2; etc.,---. What should be the position of 
the point M in order that the point M,, obtained by repeating the con- 
struction n times should coincide with M? Study the particular cases 
m=: i, 2, 3, 4. 

5. One is given an ellipse and a point P in its plane. 1° Find the 
number of circles osculating the ellipse so that each of the chords com- 
mon to the ellipse and to the different circles passes through the point. 
2° Find, for the different position of the point P, how many of these 
circles are real. 3° Prove that the points of contact of the circles oscu- 
lating the ellipse are on the circle C. 4° Find the envelope Z of these 
circles when the point P describes the ellipse. 5° The curve EF may be 
regarded as the envelope of a series of circles which intersect at right 
angles a fixed circle and whose centers lie on a conic; determine in how 
many different ways the curve # may be generated in this way. 

6. Express the co-ordinates of a point of a hyperbola as a rational 
function of a parameter ¢. What relation connects the parameters (4, te, 
tz, tg of the four points situated on acircle? How many circles osculating 


5384 PLANE GEOMETRY. BOOK IV. 


a hyperbola can be drawn through a point. ‘The circle osculating the 
hyperbola at M intersects it in M4, the osculating circle at M, intersects 
it at Mo, etc., ---; what will be the limiting position of the nth point M, 
thus constructed, when n is increased without limit ? 


7. Consider a conic C whose co-ordinates are expressed as a rational 
function of a parameter ¢ in such a way that to each point of the curve 
corresponds a single value ¢, and let A and B be two fixed points not 
situated on the conic. 

Prove that the necessary and sufficient condition that the points ¢, to, 
ts, ts of the conic C are situated on a conic passing through A and B is 


(4, — a) (t2 — a) (t3— 4) (44 — @) 
(t1 — 6) (te — b) (ts — 6) (tg — 6) 


a and 6 representing the values of ¢ which correspond to the two points 
in which the straight line AB intersects the conic C. In what way is it 
necessary to modify this relation when the straight line AB is tangent 
to C? 

8. One is given a unicursal curve of the fourth order with a triple 
point at which the tangents to the curve are distinct ; express its co-ordi- 
nates x and y asa rational function of a parameter ¢ and call a, b, ¢ the 
three values of ¢ which determine the triple point. Prove that the 
necessary and sufficient conditions that the four points ¢), te, ts, t4 situated 
on the curve lie on a straight line are: 


(t; — a)(te — a) (tz —a)(t4— a) _ (a) 
(t; — €)(t2 — ¢) (ts — c)(t4—€) $() 


(t1 — b) (te — b) (ts — db) (tg — 6) _ (0) 
(ti — €)(t2 —¢)(ts —¢)(t4—¢)  $(C)’ 


¢(t) being the common denominator of the expressions" in x and y 
Deduce the number of points of inflection and of double tangents of the 
curve. 

In what way is it necessary to modify the preceding relations in case 
two of the three tangents at the triple point coincide ? 

Apply the preceding formulas to the curve constructed in § 342 ; and 
to the curves whose equations are 


Cth, 














(x? + y?)? — ay(a? — y?) =0, 
(a? + y*)? — a(x —y)*y = 0, 
(2? + y?)? — ay® = 0. 


9. One is given a unicursal curve of the fourth order having three 
double points corresponding respectively to the values (¢=a, t=)), 
(t=a', t=b'), (t= a", t=b"). Prove that the necessary and sufficient 


CHAP. VIL. UNICURSAL CURVES. 585 


conditions in order that the four points ¢,, te, ts, t, lie on a straight line, 
take the forms j 


(1 —@ )(te—a )(tg3-—a )(te—a Tee 
(t1—b )(t2— 6 )(ts—b )(ty—b ) 


(ti — a! )(te — a! ) (ts — a! ) (tg — a!) ae t. 
(t1 — 0 )(t — 0’ )(ts — 8) (4 — 8’) 


(t, — a!') (t2 — a!') (ts — a!) (4 — !') 
(ty a, b!!) (te 2's b!') (ts = b!') (t4 hes b!’) ca 7 


equations whivh reduce to two. 

Derive the number of points of inflection and double tangents to the 
curve. 

In what way is it necessary to modify the conditions when one, two, 
or three double points become cusps ? 

Apply these formulas to the lemniscate (§ 339, 2°), to the hypocycloid 
with three cusps (curve generated by a point of a circumference which 
rolls within a circumference with a radius three times that of the rolling 
circle). 











10. Two normals which intercept between them a portion of the axis 
of the parabola of constant length are drawn to a given parabola. Find 
the locus of their point of intersection. 


11. Two parabolas whose axes include a constant angle are circum- 
scribed about a triangle: find the locus of the fourth point of intersection 
of these parabolas. (Ecole Polytechnique, 1874.) 


12. a and b are the rectilinear rectangular co-ordinates of a point J; 
what is, for each position of this point, the nature of the roots of the 
equation 

3448 at? —12bt+4+4b=—0? 

Construct, in particular, the locus of the positions of the point M for 
which the equation has a double root, and show that the co-ordinates of a 
point of the locus is a function of this root. (Ecole Normale, 1884.) 


18. Four tangents can be drawn from a point M of a lemniscate to 
this curve besides the tangent which touches the curve at M. Prove that 
the four points of contact of these tangents are on a straight line, and 
find the envelope of this straight line when the point MM describes a 
lemniscate. 

14. The envelope of the normals to a unicursal curve is a unicursal 
curve. What is the order and the class of this envelope in case of a 
conic ; a cubic unicursal curve ? 

15. One of the foci of a conic inscribed in a given triangle describes a 
given conic; prove that the other focus describes a unicursal curve of the 
fourth order. (Astor, Nouv. Annales, 1885.) 


536 PLANE GEOMETRY. BOOK IV. 


16. A straight line 6 intersects three sides of a triangle ABC in three 
points P, Q, R situated respectively on the sides BC, CA, AB. The har- 
monic conjugate P’ of P with respect to BC, Q! of @ with respect to 
CA, R' of R with respect to AB are constructed ; the three straight lines 
AP’, BQ’, CR’ intersect in a point M, 

Find the locus of this point : 

1° When the straight line 6 passes through a fixed point ; | 

2° When this straight line envelops a conic. 


e 
XX 
a 


EXAM. THE POINT.  . 537 


NOTE. 


We give below a collection of elementary examples for the application 
of the principles discussed in the text in giving an exposition of the 
theory of the point and straight line, the circle, the parabola, ellipse 
and hyperbola. 


EXAMPLES I 
CONCERNING THE POINT. 


1. Find the points whose co-ordinates are (0, 1), (— 2, 1), (— 5, 9), 
(- 2, a 3). 

2. Draw a triangle, the co-ordinates of whose angular points are 
(0, 0), (2, — 8), (—\, 0), and find the co-ordinates of the middle 
points of its sides. 

3. A straight line cuts the positive part of the axis of y at a distance 
4, and the negative part of the axis x at a distance 3 from the origin; 
find the co-ordinates of the point where the part intercepted by the axes 
is cut in the ratio 3:1, the smaller segment being adjacent to the axis 
of x. 

4. There are two points P (7, 8), and Q (4, 4); find the distance PQ, 
(1) with rectangular axes, and (2) with axes inclined at an angle of 60°. 

5. Work Ex. 4 when P is (— 2, 0), 9 (— 5, — 8). 

6. The co-ordinates of P are x=2, y=3, and of Q, e=3, y= 4; 
find the co-ordinates of R, so that PR: RQ::3: 4. 

%. The polar co-ordinates of P are p= 5, @=75°, and of Q, p=4, 
@= 15°; find the distance PQ. 

8. Find the polar co-ordinates of the points whose rectangular co- 
ordinates are 

C1) z=, (2) s=—/3, (3) x=-1, 

y= 1; y= 1; y= 1; 
and draw a figure in each case. 

9. Find the rectangular co-ordinates of the points whose polar co-ordi- 
nates are 

(1) p=5, (2) p=-5 

Oa 73 = - : = — a 


and draw a figure in each case. 


538 PLANE GEOMETRY. EXAM. 


10. Transform the equations 
xcosa+ysina=a, xz+ay+ y?= bd, 
from rectangular to polar co-ordinates. 

11. Transform the equation p? = a?cos2@ from polar to rectangular 
co-ordinates. 

12. A straight line joins the points (2, 5) and (— 2, — 3); find the 
co-ordinates of the points which divide the line into three equal parts. 

13. If ABC be a triangle, and AB, AC are taken as axes of x and y, 
find the co-ordinates (1) of the bisection of BC, (2) of the point where 
the perpendicular from A meets BC, and (8) of the point where the 
line bisecting the angle BAC meets BC. 

14. Find the co-ordinates of the same points, when AB is the axis of 
g, and a straight line drawn from A perpendicular to AB the axis of y. 

15. The rectangular co-ordinates of a point S are h and k, and a 
straight line PS is drawn, making an angle @ with the axis of x; show 
that the co-ordinates of P are 


%2=h+pcosé, y=k+psiné@, 
where SP = p. 


EXAMPLES II 
CONCERNING THE STRAIGHT LINE. 


1. Draw the lines whose equations are 


(1) y= 5a 42, (2) y-7T=52+38, (8) Ty -82=0, 
(4) 6-2 =2y, (6) 5+ = (6) 22+8=0. 


2. Find the equation to the straight line which passes through the 
points (2, 5), and (0, — 7). 

3. The co-ordinates of the angular points of a triangle being given, 
find the equations to the three straight lines, each of which bisects two 
of the sides. 

4. Two straight lines make each of them an angle of 45° with the 
axis of z, and their intercepts on the axis of y are 6 and 8; find the 
equation to the straight line which is equidistant from the two, on 
axes being rectangular. 

5. Find the equation to a straight line on which the perpendicular 
from the origin = 6, and makes (1) an angle of 45°, and (2) an ee 
of 225° with the axis of x, the axes being rectangular. 

6. Determine the point of intersection of the two lines (3 y —x=0) 
and (2%+ y=1). 


EXAM. THE STRAIGHT LINE. 539 


%. Find the equation to the straight line which passes through the 
point of intersection of the straight lines 


x—2y—a=0, ++ 38y-—2a=0, 
and is parallel to the line 8%+4 4y =0. 


8. Find the equation to a straight line which is equidistant from the 
two lines represented by the equation y= mz +ce+4c'. 

9. Find the equation to the straight line that joins the points of inter- 
section of the two pairs of lines 


24+ 3y—-—4a=0, and “+ 8y—7a=9, \ 
2e4+y—a=)0, Apes eo, 

10. Find the length of the perpendicular from the origin on the line 
a(x —a)+ b(y —b) =0, and the portion intercepted by the axes, which 
are rectangular. 

11. The rectangular co-ordinates of two points are 3, 5 and 4, 4 
respectively ; find the equation to a straight line which bisects the dis- 
tance between them and makes an angle of 45° with the axis of x. 


12. Find the equation to a straight line which passes through a given 
point (a, b) and makes equal angles with the axes. 


13. Find the equations to the diagonals of the parallelogram formed by 
the four lines x =a, x=a'/,y=b, y= 0'. 

14. A straight line, inclined to the axis of x at an angle of 150°, cuts 
the positive axes of rectangular co-ordinates in A and B; find the equa- 
tion to a straight line bisecting AB and passing through ne origin. 

15. Find the equations to the four sides of a square, the co-ordinates 
of two of its opposite angular points being (2, 3) and (38, 4), the co-ordi- 
nates being rectangular. 

16. Find the distance of the origin of co-ordinates from the line 

y 
Se 

17. Find the equation to a straight line which passes through the inter- 
section of the lines x = a, x + y + a =0, and through the origin. 


_ = 1, the axes being rectangular. 


18. The axes of co-ordinates being inclined to each other at an angle 
of 60°, find the equation to a straight line parallel to the line («7 + y = 3a) 
and a distance from it equal to Lav3. 

19. Show that the lines y= 2243, y=3x+4+4, y=4a + 5, all pass 
through one point. 

20. Find the value of m, in order that the line (y = mx + 3) may pass 
through the intersection of the lines (y=x+1) and (y=2% +4 2). 

21. A straight line cuts off intercepts on the axes, the sum of the 
reciprocals of which is a constant quantity ; show that all straight lines 
which fulfill this condition pass through a fixed point. 


540 PLANE GEOMETRY. , EXAM. 


22. A straight line slides along axes of x and y, and the difference of 
the intercepts is always proportional to the area it incloses ; show that the 
line always passes through a fixed point. 


23. If the distance of a point from the origin equals twice its distance 
from the axis of x, show that it always lies in one of two straight lines 
that pass through the origin ; axes rectangular. 


24. Find the cosine of the angle which the line (Az + By + C) makes 
with the axis of x, the axes being inclined at an angle of 45°. 


25. If a straight line cuts the (rectangular) axes of x and y at equal 
distances from the origin, and a straight line be drawn from the origin, 
dividing it in the ratio m:n, find the tangent of the angle which this latter 
line makes with the axis of . 


26. An equilateral triangle, whose side =a, has its vertex at the 
origin, and its sides equally inclined to the positive directions of rectan- 
gular axes; find the co-ordinates of the angles, and thence of the point 
bisecting the base. 


27. Find the polar co-ordinates of the point of intersection of the lines 


p=2asec(o—™ \ and { =—asec(o— \ 
{ Pp ( 5) ) p 6 ’ 
and the angle between them. 
28. Trace the line whose polar equation is 
| p=2acos (9+): 
6 
29. Show that the polar equation to a straight line, passing through 
the points (p', 0’), (p", @/’), is 
p'p sin (6! — 0) + p!'p! sin (0"’ — 6’) + pp!’ sin (8 — 6!) = 0. 


What is the geometrical interpretation of this equation? -‘ 


EXAMPLES III 


CONCERNING THE ANGLES FORMED BY STRAIGHT LINES AND 
THE STRAIGHT LINES REPRESENTED BY EQUATIONS OF 
SECOND AND HIGHER DEGREES. 

1. Find the equation to the straight lines which pass through the point 


(1, 3), and make an angle of 30° with the line (2y—x%+4+1=0); axes 
being rectangular. 


2. Draw the lines represented by the equation 
(Qy—x+ce)8y+x-—c)=9, 
and determine (1) where they intersect, and (2) at what angle; the axes 
being rectangular. 


EXAM. TRANSFORMATION OF CO-ORDINATES 541 


3. Find the equation toja straight line which passes through the point 
(c, 0), and makes an angle of 45° with the line (bx —ay= ab); axes 
being rectangular. 


4. Find the equation to a straight liné which is perpendicular to the 
line (8y+52—38=0) and cuts the axis of y at a distance = 8 from the 
origin ; axes being rectangular. 

5. Find the cosine of the angle between the lines 

(y—4x+8=0) and (y—6%+9=0); 
axes being rectangular. 

6. Find the angle between the lines 

(4y+82+5=0) and 4x%—38y7+4+6=0); 
axes being rectangular. 

7%. Find the equations to the straight lines which pass through the 
intersection of the lines (y=2%+4), (y=38x+6), and bisect the 
supplementary angles between them ; axes being rectangular. 

8. What is the geometrical signification of the equations 

: x2 + y2 = 0, | eet A 

9. Find the equations to the straight lines which bisect the angles 
between the lines (12%+5y=8) and (8}%—4y=8); axes being 
rectangular. 

10. Show that the lines represented by the equation 
6y2+ay—2ae+y—x2-—-1=0, 
are inclined to one another at an angle of 45°; axes being rectangular. 
11. The equation 27? —3ay—222—38y+6x2=0 represents two 
straight lines at right angles ; axes being rectangular. 
12. The equation y? — 2 xy sec @ + x2 = 0 represents two straight lines 
inclined to one another at an angle @; axes being rectangular. 
13. What is the inclination of the co-ordinate axes, when the lines 
represented by y? — x? =0, are perpendicular to one another ? 
14. The equations to two straight lines are 
r+3sy—a=0---(1), y—“2+a=0-+- (2); 
find the equations to the straight lines which pass through the intersec- 


tions of (1) and (2), so that the ratio of the sines of the inclination of 
each to (1) and (2) may be as 1: V5. 

15. What must be the inclination of the axes in order that the lines 
(xy —38y —2%+6=0) may include an angle of 135° ? 

16. Find the equations to the two straight lines which pass through 
the origin and divide into three equal parts the distance between the 
points in which the axes of co-ordinates are intersected by the line 
(«+y=1). 


§42 PLANE GEOMETRY. EXAM. 


%. Find the distance of the point of intersection of the lines 
(3a+2y+4=0), (22+5y+8=0), 


from the line (y = 5a + 6) ; the axes being rectangular. 


EXAMPLES IV 
CONCERNING TRANSFORMATION OF CO-ORDINATES. 


1. The equation of a right line is 
82+ 5y—15=0; 


find the equation of the same line referred to parallel axes whose origin 
is at (1, 2). Ans. 8x+ 5y =2. 


2. The equation of a locus is 
e+y?—4x-—6y = 18; 


what will this equation become if the origin be moved to the point 
(2, 3)? Ans. 224+ y?=81. 


3. The equation of a locus is y? — #?= 16; what will this equation 
become if transformed to axes bisecting the angles between the given 
axes ? Ans. TY =o, 


4. Transform the equation 2 «2 — 5ay + 2y? = 4 from axes inclined to 
each other at an angle of 60°, to the axes which bisect the angles between 
the given axes. _ Ans. x? — 27y2+4+12=0. 


5. Transform the equation y? + 4aycota —4ax=0 from a rectan- 
gular system to an oblique system inclined at an angle’ a, the origin 
remaining the same, and the new axis of « coinciding with the old. 

Ans. y?sin?a = 4 ax. 


6. The equation of a locus is xt + y* + 6a2y? = 2; what will be the 
equation if the axes are turned through an angle of 45°? 
Ans. x§+ yt =1. 


”. Transform x? + y2 = 7 ax to polar co-ordinates, the pole being at the 
origin, and the initial line coincident with the axis of «. 
Ans. r=7acosé. - 


8. Change the equations r? = a? cos26 and 72 cos26 = a? into equa- 
tions between andy. Ans. (x? + y?)? = a? (a? — y?) and 2? — y? =a’. 


EXAM. THE CIRCLE. 543 


EXAMPLES V 
CONCERNING THE CIRCLE. 


1. To find the center and radius of the circle 
e+ y?—644+4y+4=0. 
2. Investigate the line or lines represented by the equation 
x8 + ay? — 22 — er? — ry? + 3 = 0. 
3. Find the common chord of two circles 
(4@—-1)?+(y—2)?=6, @—2)?+(y— 3)? =8. 


4. To find the equation to a straight line which passes through the 
centers of the two circles 


e+2¢+y=0, y2+2y4+27=0. 


5. To find the equation to a circle having for its diameter the straight 
line joining the points of intersection of the line y = mz and the circle 
y= 2re — 2%. 


6. Find the equation to the circle the diameter of which is the com- 
mon chord of the circles 


w+ yar, (2—alt+y2=r. 
7. What is represented by the equation 
x(x—2)+y(y—4)+8=0? 
8. Find a relation between the coefficients of the equation 
A (2? + y2)+ Dx + Ey + F=0, 


in order that (1) the axis of x, and (2) the axis of y, may be tangents to 
the circle. 


9. To find the inclination to the axis of x of the tangents drawn from 
any point («’, y’) to the circle whose equation is 
(1—a)?+(y— b)?-7r?=0. 
10. To find the relation between the quantities a, b, r, in order that 
the line “ a ; = 1 may touch the circle x? + y? = r?. 


11. To find the equation to a circle, the center of which is at the origin 
of co-ordinates, and which is touched by the line 


12. To find the intercepts on the axes of co-ordinates of the tangent 
to a circle (x? + y? = r?), drawn parallel to a given straight line 


(xcosa+ysina =p). 


544 PLANE GEOMETRY. [sae 


13. If 2a’, 2a! be the inclination of two radii of a circle x2 + y2 = 1? 
to the axis of x, to find the equation to the chord joining the extremities 
of the radii. 


14. If the pole always lie on a line 
ag Ae 
ab 


and the equation to the circle is x? + y? = r?, the equation to the polar is 


of the form 
(ax —r?)+ k (by — 7?) =0, 


where k is any constant. 


15. If the pole of a straight line with regard to the circle 27+ y? = v2 
lie on the circle x? + y? = 4r?, the polar will touch the circle 


x2 ee Sean 
ee 4 


16. Find the equation to the circle which has each of ‘the co-ordinates 
of the center = — A and its radius = ae the axes being inclined at an 
angle of 60°. 

17. Prove that the circles 

a+ y2=(c+a)%, @—a)?+y=e 
have only one common tangent, and find its equation. 

18. Find the locus of the mid-points of chords drawn from the ex- 
tremity of the diameter of any circle. 

19. Show that the polar of the point («’, y’) with regard to the circle 
(x — a)? +(y — b)?=7°" is 

(a —a)(a! —a)+(y—b)(y'—b) =”. 

90. Find the locus of the vertices of all triangles which" have a given 
base and a given vertical angle. 

21. Prove Euc. III. 31, from the resulting equation. 


22. Tangents are drawn to a circle 27+ y?= 72, at two points (x, y'), 
(a!', y!’); to find the distance of a point (h, k) from a straight line pass- 
ing through the center and the intersection of the two tangents. 


93. To find the equations to straight lines touching a circle 
a+ y? = 10, 
at points, the common abscissa of which is unity. 
94. Find the equation to a straight line touching the circle 
| (27—a)?+(y— b=, 
and parallel to a given line y = mx + ¢. 


EXAM. THE CIRCLE. — 545 


25. To find the equation to the straight line passing through the origin 
of co-ordinates, and touching the circle 
g2+y2?—38xe4+4y=0. 
26. To find the length of the common chord of the circles 
(a —a)?+(y—b)2 =r, (@— bd)? +(y— a)? =P 

27. Find the area between the two circles 

24+2e+y2+4y=—0, v2 +2e+y2?+4y=1. 
28. To find the length of the chord of a circle x? + y2 = 1r?, made by 


the straight line : + 4 <a 


29. If from a given point S, a perpendicular be drawn to the tangent 
PY at any point P of a circle, of which the center is C, and, in the line 
MP at right angles to C'S and produced if necessary, a point @ be taken, 
such that QM = SY, to find the locus of Q. 


30. Given the equation to a circle, and the chord of a circle; show 
that a perpendicular let fall upon the chord from the center bisects the 
chord. 


$1. Find the diameter of the circle 
2 + y2+ 2xy cosw = ax + by. 
32. In the equation Ax + By + C=0, if C be constant, and A and B 


vary, subject to the condition A? + B? =a constant, the equation repre- 
sents a series of tangents to a given circle. 


33. Find the equation to the circle which passes through the points 
(0, 0), (— 8a, 0), (0, 6a), the axes being rectangular. 

34. To find the locus of mid-points of chords which pass through a 
given point. 

35. If on any radius vector through a fixed point O, O@ be taken ina 
constant ratio to OP, find the locus of Q. 

36. The circles represented by the equation 

(n +1) (a? + y?) = ax + nby, 
where n is arbitrary, have a common chord. 

37. Prove algebraically that the angles in the same segment of a circle 
are equal, and that the angle in a semicircle is a right angle. 

38. Two sides of a triangle are } and c, and they include an angle A; 
if these sides be taken as axes, the equation to the circumscribed circle is 
x? + y2 + 2x2y cos A — ba — cy = 0. 

39. Given the base and vertical angle, to show that the locus of the 
point of intersection of the perpendiculars from the angles on the sides is 
a circle. 

2M 


546 PLANE GEOMETRY. | EXAM. 


40. Given base and ratio of sides of a triangle, show that the locus of 
the vertex is a circle. 


41. When will the locus of a point be a circle, if the square of its dis- 
tance from the base of a triangle be in a constant ratio to the product of 
its distances from the sides ? 


42. When will the locus of a point be a circle, if the sum of the 
squares of the three perpendiculars from it on the sides of a triangle be 
constant ? 


43. Find the locus of a point, the square of whose distance from a 
given point is proportional to its distance from a given right line. 


44. Given the base of a triangle, and m times the square of one of its 
sides + » times the square of the other =a constant ; find locus of the 
vertex, find center and radius of resulting circle, and where it cuts base. 


45. Find the equations to the circles which touch the three lines, re- 
ferred to rectangular axes, 


C=O = 20, ee 0 
46. The locus of the centers of all circles inscribed in all right-angled 
triangles on the same hypotenuse is the quadrant described on the 
hypotenuse. 
47. The equation to acircle is y? + «?=a(y+ 2x); what is the equa- 
tion to that diameter which passes through the origin of co-ordinates ? 


48. To find the equation to a circle referred to two tangents at right 
angles, as axes. 


49. If through any point of a quadrant whose radius is & two circles 
be drawn touching the bounding radii of the quadrant, and r, r’ be the 
radii of these circles, r7’ = R?. 


50. To find the equations to the straight lines which touch both the 

circles 
42 + y? —_ r2, (x fe a)? + y? — y!2, 

‘51. To find the equation to the circle which touches the three straight 

lines, referred to rectangular axes, 
‘ce 
=O, .9 = 0, es 

52. To find the equations to two circles, which touch rectangular axes 
of x and y, and pass through a given point (a, b). 

53. The straight lines joining the angles of a triangle with the points 
in which the escribed circles touch the opposite sides, meet in a point. 


54. In any circle draw a chord AB; from the mid-point # of the 
lesser segment draw any straight line cutting AB in C, and meeting the 
circumference in D; join AD, and in AD take AP= AC; find locus 
of P. 


| EXAM. LOCI OF THE SECOND DEGREE. -  §4T 


55. The axes Ox, Oy cut a circle in points A, A’, B, B’ respectively ; 
to compare the values of x, y at the intersection of the chords AB’, A'B. 


56. Determine the magnitude and position of the circle 


p2 — 2 p(cos 6+ V3 sin 6) = 5. 


EXAMPLES VI 

Transform the following equation, illustrating each transformation 
by a figure, as at the end of § 144. 

v1. 14224+3y=0 to Y=— x. 
8224 2y2?-2Qet+y—1=0 to 7222 + 48 y? = 35. 
8424 2ay+3y2 —16y+ 23=0 to 42242y=1. 
y2?—10ay + a2 +y4+2441=0 to 822? — 48y7=9. 
y? —2ay + a2 —6y—624+9=0 to y2=3 V2. 
yretaytat+ty+2—5=0 to 9224 37? = 82. 
y? —2ay —22+2=0 to y2—22?4+V2=0. 
y2?—a2—y=0 to 422?-4y?4+1=0. 


oD TOT FP wo 


Show by transformation that the equation 
12ay+ 8a%—27y—18=0 
represents two straight lines parallel to the axes. 
10. Show by transformation that the equation 
y? —2ey+3a?-2y—10x%4+19=0 
represents two imaginary straight lines passing through the point (8, 4). 
11. Show by transformation that the equation 
y2—4aey+5a2?+2y—4”474+2=0 
represents an imaginary ellipse. 


12. Show that any point on the line (y=%+1) is a center of the 
locus 
y2 —2Qeyt+u?—2y+2x=0. 


138. Show by transformation that the equation 
a, y2+2ey+e74+1=0 
represents two imaginary parallel straight lines. 
14. What is the equation to the axis in Ex. 5? 


15. Transform. 7 y? + 16 zy + 16224 382y+ 64%+428=0, the axes 
being inclined at an angle of 60°, to y2+422=9, the axes being rec- 
tangular, and the axis of « remaining the same. 


548 PLANE GEOMETRY. EXAM. 


EXAMPLES VII 
CONCERNING THE PARABOLA. 


1. Find the intersections of the parabola y?=8~z and the line 
dy—2x—8=0. Ans. (2,.4) and (8, 8). 
2. Find the equation of the right line passing through the focus of the 
parabola y2 = 4%, and making an angle of 45° with the axis of the curve. 
Ans. y=x-1.- 

8. Find the points in which the focal chord y= # — 1 intersects the 
parabola y? = 42. ‘Ans. (84 2V2,.2 4 2v2). 
4. Find the equation of the right line passing through the vertex of 
any parabola and the extremity of the focal ordinate. Ans. Y= 2s. 


5. Find the equation of the circle which passes through the vertex of 
any parabola and the extremities of the double ordinate through the focus. 
Ans. y? = 3 px — x. 
6. Find the equation of the circle which passes through the vertex of 
the parabola y? = 12 x and the extremities of the double ordinate through 
the focus. . Ans. y? = 15% — x?. 
7. Find the equations of the tangent and normal to any parabola at 
the extremity of the positive ordinate through the focus. 
Ans. y=u4+4p andy+2= 8p. 
8. Find the equations of the tangent and normal to the parabola 
y? = 4x, at the extremity of the positive ordinate through the focus. 
Ans y=2+135 94+ 2=8. 
9. Find the point where the normal in Ex. 7 meets the curve again, 
and the length of the intercepted chord. 
Ans. (3p, —3>p); length of chord = 4 pv2. 
10. Find the point where the normal in Ex. 8 meets the curve again, 
and the length of the intercepted chord. 
Ans. (9, — 6); length of chord = 8v2. 
11. Find the point in a parabola where the tangent is inclined at an 
angle of 30° to the axis of x. Ans. (3p, pv3). 


12. Prove that the normal at any point of a parabola bisects the angle 
between the focal line and the diameter passing through that point. 


13. On a parabola whose latus rectum is 10, a tangent is drawn at the 
point whose ordinate is 6, the origin being at the principal vertex ; deter- 
mine where the tangent cuts the two co-ordinate axes. 

Ans. (— 38.6, 0) and (0, 38). 

14. Determine where the normal in the preceding example, at the 


same point, if produced, will cut the two axes. 
Ans. (8.6, 0) and (0, 10.3). 


EXAM. THE PARABOLA. 549 


15. Find the angle which the tangent in Ex. 14 makes with the axis 


of x. Ans. 39° 48! 20". 
16. In the parabola y? = 122, find the length of the perpendicular from 
the focus to the tangent at the point whose abscissa is 9. Ans. 6. 
17. In the parabola y? = 82, find the length of the normal at the point 
whose abscissa is 6. Ans. 8. 


18. The extremities of any chord of a parabola being (a!, y’), (2, y!), 
and the abscissa of its intersection with the axis of the curve being «, to 
prove that y!x!! = x, yy!’ =— 2 px. 

19. Two tangents of a parabola meet the curve in (a’, y’) and («!, y’’), 
their point of intersection being (#, y); show that 


g=Valc", y= os 


EXAMPLES VIII 


The following problems are enunciated, some for the ellipse and some 
for the hyperbola, though many of them are equally applicable to both 
curves. 


1. Find the semi-axes of the ellipse 3 y? + 2 #? = 6. 
Comparing this equation with - + a = 1, we find 
a 
a—v3, and b = V2, Ans. 


9. Find the semi-axes of the ellipse 4 y? + 3a2 = 19. 
Ans. a=vif, b=Vi2. 


3. Find the points of intersection of the parabola y?7=4% and the 


ellipse 3 y? + 24? = 14. Ans. (1, 2) and (1, — 2). 
4. Find the equation of a tangent to the ellipse 3 y? + 2”? = 35, at the 
point whose abscissa is 2. Ans. 9y+4a =35. 


5. Find the eccentricity of the ellipse 2 22 + 3 y? = d?. 
Ans. Eccentricity = V2. 


6. Find the equation of the tangent to the ellipse at the end of the 
latus rectum ; also, find the lengths of the intercepts of this tangent on 
the two axes. 

Ans. y+ex =a; the intercepts are “ on the axis of 2, and a on the 
axis of y. ‘ 

”. Find the axes of the hyperbola whose equation is 8 y?—22?+12=0; 


also the eccentricity of the given and the conjugate hyperbola and the 


parameter. Ans. a=V6, b=2; e=V3; d=Vv3; So 


v6 


550 PLANE GEOMETRY. EXAM. 


8. Find the intersection of the hyperbola 3 y? — 2x? + 12 = 0 and the 


circle x? + y? = 16. Ans. (4 2V3, 12). 
9. Find whether the line y = 3 x cuts the hyperbola 5 y? —2 a2 =— 15, 
or its conjugate. Ans. It cuts the conjugate. 


10. Find the equation of an hyperbola of given transverse axis, whose 
vertex bisects the distance between the center and the focus. 


Ans. y2 —- 3822 =— 3 a?. 
11. Show tan P. SH a an PHS _1— | where P is any point on an ellipse 


2 ey 





and H and S are its foci. 

12. Find the points of intersection of an ellipse and hyperbola whose 
equations are #2+2y?=1, 322—6y?=1, and show that at each of 
these points the tangent to the ellipse is the normal to the hyperbola. 


13. If CA, CB be the semi-axes of an ellipse, show that, when SBH is 
a right angle, CA?: CB? =2: 1. 


14. Find the condition that the line (2 + ie .): should touch the 
2 2 7, le 
hyperbola (s Ya 
a2 


D4 


« 


15. The tangent to an ellipse is inclined to the major axis at an angle 
¢@ ; show that the area included by this tangent and the axes is 


= } (a? tan ¢ + b2 cot ¢). 
16. The circle described on any radius vector SP of an ellipse as 
diameter will touch the circle on the axis major. 


17. Find where the tangents from the foot of the directrix will meet 
the hyperbola, and what angle they will make with the transverse axis. 
18. Find the equation to the tangent at the extremity of the latus 
agi 
Mgt ae 
19. A tangent at the extremity of the latus rectum of an hyperbola 
meets any ordinate PM produced in R; show that SP = MR, where S is 
the focus through which the latus rectum passes. 


ft 








20. Show that the equation to the normal, at the point whose eccentric 
angle is ¢, is ax sec @ — by cosec @ = a® — b*. 

91. Find the radius of a circle inscribed in a semi-ellipse, touching the 
axis minor. 

92. From the point where the circle on the major axis is intersected by 
the minor axis produced, a tangent is drawn to the ellipse; find the point 
of contact. 

93. If from the extremities of the minor axis two straight lines be | 
drawn through any point in the ellipse, and intersect the axis majorin @ 
and R, then CQ. CR =CA?. 


EXAM. ELLIPSE AND HYPERBOLA. | 551 


94. If a tangent be drawn to the interior of two concentric ellipses, 
the axes of which are in the same straight line, meeting the exterior one 
in P, Q and at P, Q tangents be drawn to the latter, intersecting in R, 
prove that the locus of J? is an ellipse. 


95. Show that the locus of one end of a given straight line, whose 
other end and a given point in it move in straight lines*at right angles to ~ 
one another, is an ellipse. 


96. If with the co-ordinates of any point in an elliptic quadrant as 
semi-axes, a concentric ellipse be described, the chord of the quadrant of 
the one will be a tangent to the other. 


97. The locus of the center of a circle touching two circles externally 
is an hyperbola. Z 


98. The locus of the center of a circle touched by one circle externally, 
and one internally, is an hyperbola. 


99. Find the locus of the extremity of the perpendicular from the 
center on the tangent to the hyperbola. 


30. If 6, @ be the eccentric angles of two points on an ellipse, the 
equation to the chord joining the two points is 


Spee oe 4 ogee _ POR Sota ce 
a 2 b 2 2 


Hence deduce the equation to the tangent at the point whose eccentric 
angle is @. 

31. If 83 4C =2CS8 in an hyperbola, find the inclination of the asymp- 
totes to the transverse axis, 


32. If from a point P in an hyperbola, PX be drawn parallel to the 
transverse axis, cutting the asymptotes in J and K, then Bo « GEae tl oF 
or, if parallel to the conjugate, PK - PI = b?. 


33. Is the point (2, 3) without or within the hyperbola 222 —3y?=7? 
Show that the straight line, joining this point with the point (6, 4), cuts 
the curve. 


34. If A, A’ be the extremities of the major axis of an ellipse, T the 
point where the tangent at P meets AA’, QTR a line- perpendicular to 
AA!, and meeting AP, A'P in Q and BR respectively, then QT = TR. 


35. Find the eccentricity and latus rectum of the conic 
Qy2+ar2+4y—2xe-6=0, 


the axes being rectangular. 


552, PLANE GEOMETRY. EXAM. 


86. Find the equations to the asymptotes of the curve 
3x2? — 10x%y + 3y2?—8=0; 
and find the angle between the asymptotes of 
y?—1l0ay+a7+y7+244+1=0, 


the axes in the latter case being rectangular. 


87. In the equilateral hyperbola, the eccentricity is the ratio of the 
diagonal of a square to its side. 


38. A tangent at any point P of an ellipse meets the axis major pro- 
duced in 7, and the axis minor produced in ¢; to find the locus of a point 
@ in 7¢ such that QT’: Qt =m: n. 


39. To find the locus of the intersection of the ordinate of any point in 
an ellipse produced with the perpendicular from the center upon the 
tangent at that point. 


40. If the normal at P meet the axis major of an ellipse in G, and GK 
be drawn perpendicular to SP, GA =e - PM, where PM is the ordinate 
of Ff. 


41. If SQ be drawn, always bisecting the angle PSC, in an ellipse, 
and equal to a mean proportional between SC and SP, find the eccen- 
tricity of the curve which is the locus of Q. 


42. Two straight lines, such that the product of the tangents of their 
inclinations to the axis of x is constant, touch an ellipse ; show that the 
locus of their intersection is. an ellipse, or hyperbola, according as the 
product is negative or positive. 


so 

48. Show that the locus of the summit of a movable right angle, one 

side of which touches one, and the other side the other, of two confocal 
ellipses, is a concentric circle. 


44. An ellipse and hyperbola have the same foci and coincident axes ; 
they cut each other at right angles. 


45. If P be any point in the hyperbola, S and H the foci, find the locus 
of the center of the circle which is inscribed in SPH. 


46. If a tangent at any point of an hyperbola be intersected by the 
tangents at the vertices in H and K, the circle on HK as diameter passes 
through the foci. 


oF 


EXAM. - CONJUGATE DIAMETERS. 553 


EXAMPLES IX 


CONCERNING THE ELLIPSE, HYPERBOLA, AND THEIR CONJUGATE 
DIAMETERS. 


1. If CP, CQ be semi-diameters at right angles to each other, 


1 | aodete See oo 
CP? CQ a? Bb? 
2. If, from the focus S of an ellipse, perpendiculars be drawn on CP, 
CD conjugate diameters, these perpendiculars produced backwards will 
intersect CD and CP in the directrix. 


8. If p, rv and p!,r’ be respectively the focal distances of two points P, 
D, the extremities of a pair of conjugate diameters of an ellipse, then 


pr + pir! = a? + 0. 
4. If a tangent to an hyperbola at P cut off C7, Ct from the axes, 
then PT. Pt = CD?, CD, being the semi-conjugate diameter. 


5. In the equilateral hyperbola, the conjugate diameters make equal 
angles with the asymptotes. 


6. From the extremities P, D of two conjugate diameters, normals are 
drawn to the major axis of an ellipse; the sum of the squares of these 


b2 
two = —(a? + b?). 
a? ; 


7. If the tangent at the vertex A. cut any two conjugate diameters of 
an ellipse produced in 7 and ¢, then AZ’. At = b?. 


8. The lengths of the equal conjugate diameters of an ellipse are 
V2(a? + b?), and the eccentric angles of their extremities are 45° and 
185°. 


9. The locus of the mid-points of chords of an ellipse, which pass 
through a fixed point, is an ellipse with the same eccentricity ; and if the 
fixed point be the focus, the major axis of the ellipse is SC. 


10. The tangent at any point of an hyperbola is produced to meet the 
asymptotes; show that the triangle cut off is of constant area. 


11. If the asymptotes of the hyperbola are axes, show that the equation 
‘to one directrix is x+y—a=0. 


12. If any two tangents be drawn to an hyperbola, and their intersec- 
tions with the asymptotes be joined, the joining lines will be parallel. 


13. Show that the locus of the points of quadrisection of all parallel 
chords in a circle is a concentric ellipse. 


554 PLANE GEOMETRY. EXAM. 


14. If the angle between the equal conjugate diameters of an ellipse is 
60°, find the eccentricity. 


15. If a be the angle between two conjugate diameters which make 
angles 6, 6’ with the axis major, 


cosa = e2 cos 6cos 6!. 


16. CP, CD are semi-conjugate diameters of an ellipse, and PF is a 
perpendicular let fall from P on CD or CD produced; determine the 
locus of F. 


17. The chords joining the extremities of the conjugate diameters of 
an ellipse will all touch, in their mid-points, a concentric ellipse with 
axes av2, bV2 coincident with those of the original curve. 


18. If acircle be described from the focus of an hyperbola, with radius 
equal to half the conjugate axis, it will touch the asymptotes in the points 
where they are cut by the directrix. 


19. Trace the curve, referred to rectangular axes, 
a ler 2 
(4a 89) CEES BX a8 


90. The radius of a circle, which touches an hyperbola and its asymp- 
totes, is equal to that part of the latus rectum produced, which is inter- 
cepted between the curve and the asymptote. 








21. The equation to the diameter conjugate to 
tY is — +20, 
cs 5 6 Re 
the hyperbola being referred to its asymptotes, c and s are the cos and 
sin of the angle formed by the line and the “-axis. 
of 

92. An ellipse being traced upon a plane, draw the axes and the direc- 

trix, and find the focus. 


93. Find the angle between the asymptotes of the hyperbola xy=b2? +¢, 
the axes being rectangular; and write the equation to the conjugate 
hyperbola. 


24. Tangents are drawn to an hyperbola, and the portions intercepted 
by the asymptotes are divided in a given ratio; show that the locus of 
the point of division is an hyperbola. 


25. Draw the asymptotes of the hyperbolas 
ry —2"4—3y—2=0, zy +22?4+3=0, 
and place the curves in the proper angles. 


96. Find the locus of the intersection of tangents to an ellipse, which 
are parallel to conjugate diameters. 


EXAM. CONJUGATE DIAMETERS. 555 


27. Find the equation to the locus of the mid-points of all chords of a 
given length, in an ellipse. 


28. If two concentric equilateral hyperbolas be described, the axes of 
the one being the asymptotes of the other, they will intersect at right 
angles. 


29. If P be the mid-point of a straight line AB, which is so drawn 
as to cut off a constant area from the corner of a square, its locus is an 
equilateral hyperbola. 


30. If Sand H be the foci of an equilateral hyperbola, and a circle be 
described upon S//, then the quadrantal chord of this circle shall be a ~ 
tangent to that described upon the transverse axis. 


31. If a be the acute angle between the axes of co-ordinates of the 
ellipse (2? + y? = c?), find the lengths of the axes and the eccentricity. 


32. If AA!’ be any diameter of a circle, PQ any ordinate to it, then the 
locus of the intersections of AP, A’Q is an equilateral hyperbola. 


33. In an equilateral hyperbola, focal chords parallel to conjugate diam- 
eters are equal. ; 


34. If a series of straight lines have their extremities in two straight 
lines at right angles to one another, and all pass through a given point, 
the locus of their mid-points is an equilateral hyperbola. 


35. PQ is an ordinate to the axis major AA’ of an ellipse, meeting the 
curve in Pand @; draw AP, A’Q intersecting in R; the locus of R is an 
hyperbola with the same center and axes. 


36. If tangents be drawn, making a given angle with the axes of all 
ellipses having the same foci, the locus of the point of contact is an 
equilateral hyperbola. 


37. If normals be drawn to an ellipse from a given point within it, the 
points where they meet the curve will all lie in an equilateral hyperbola 
which passes through the given point, and has its asymptotes parallel to 
the axes of the ellipse. 


38. Find the locus of the mid-points of chords in a circle, which 
touch a concentric ellipse. 


39. If normals be drawn from the extremities of conjugate diameters 
to an hyperbola, and the point of their intersection be joined to the 
center, this line produced shall be perpendicular to the straight line pass- 
ing through the extremities of the conjugate diameters. 


40. Given in position, a straight line AB and a point P outside it; a 
straight line PM is drawn, intersecting AB in C, from the extremity 
of which a perpendicular MD on AB intercepts CD of a given magni- 
tude ; find the locus of . 


556 PLANE GEOMETRY. EXAM. 


41. The locus of the centers of all circles, which cut off from the direc- 
tions of two sides of a triangle chords equal to two given straight lines, is 
an equilateral hyperbola, having two conjugate diameters in the direc- 
tions of these sides. 


42. A straight line passes through a given point and is terminated in 
the sides of a given angle ; find the locus of the point which divides it in 
a given ratio. 


43. From a point P perpendiculars are dropped upon the sides of a 
given angle, so as to contain a quadrilateral of given area; show that the 
locus of P is an hyperbola whose center is the vertex of the given angle. 

44. Given the base of a triangle and the difference of the tangents of 
the base angles; show that the locus of the vertex is an hyperbola, of 
which the perpendicular through the center of the base is an asymptote. 


45. If about the exterior focus of an hyperbola, a circle be described 
with radius equal to the semi-conjugate axis, and tangents be drawn to it 
from any point in the hyperbola, the straight line joining the points of 
contact will touch the circle described on the transverse axis as diameter. 


46. If, from the center of an equilateral hyperbola, a straight line be 
drawn through any point P, and if ¢ and ¢’ be the angles which this line 
and the polar of P respectively make with the transverse axis, then 


tan ¢ tan ¢/ = 1. 
47. Prove that the circle which passes through any three of the four 
points in which the equilateral hyperbola 
y2+2hey —y?+29x%4+2fy+c=0 
cuts the rectangular co-ordinate axes, is equal to the circle 
ge + y2+2gx+2 fy =—0. 
48. Find the locus of the mid-points of a system of parallel chords, 
drawn between an hyperbola and the conjugate hyperbola. 


ft 


49. If, in two concentric hyperbolas, whose axes are coincident, two 
points be taken whose abscissas are as the transverse axes of the hyper- 
bolas, the locus of the mid-point of the straight line joining them is an 
hyperbola, whose axes are arithmetical means between those of the given 
hyperbolas. 


50. If tangents be drawn from different points of an ellipse, of lengths 
equal to n times the semi-conjugate diameter at the point, the locus of 
their extremities will be a concentric ellipse with semi-axes equal to 
avn? +1, bVn? +1. 


51. If a length PQ = CD be taken in the normal to an ellipse, the 
locus of the point Q is a circle whose radius = a — b or a + b, according 
as Q is taken within or without the ellipse. 





EXAMINATION QUESTIONS. 55T 


QUESTIONS PROPOSED FOR VARIOUS 
EXAMINATIONS 


ECOLE POLYTECHNIQUE. 


1860. A parabola P is given; let A and B be two variable points on 
this curve, but so chosen that the normals at A and B intersect in a point 
C situated on P. The tangent at C and the straight line AB intersect in 
a point N; find the locus described by this point and construct the curve. 


1863. Two circumferences O, O’ are given in a plane; from a point 
A situated on O, tangents are drawn to O’, and the points thus deter- 
mined are joined. This straight line intersects the tangent at O, at the 
point A, in MW; find the locus described by M. 

Investigate the different forms of this locus according to the relative 
magnitudes and position of the two circumferences O and 0’; indicate 
the cases in which the locus is decomposable. Show that the locus of 
the points M is tangent to O at each of the points which are common to 


- it and this circumference. 4 


1864. Construct the circle whose equation is 





2 + y? —_ 1, 
and the parabola which has the equation 
3 a2 2_] 
(Bx — ay)? + 2a” + 2 By = — ; 


where a and f are any positive parameters. It is proposed to determine: 
1° The number of real points common to the two curves, for the differ- 
ent values of a and £. 
2° The co-ordinates of the four common points when 


at + 6? = 1, 
or a=1, and B>0; 
finally, when we have 





B=V (a? —1)(4a?—1). 


1865. A parabola P is given in a plane and a circumference C pass- 
ing through the focus P is considered. It is required to find the regions 
of the plane which the center C will occupy in order that this circumfer- 
ence may have successively in common with P: 1° four real points ; 


558 PLANE GEOMETRY. 


2° four imaginary points; 3° two real points and two imaginary points. 
Study the form and the properties of the curve which separate the first 
two regions from the third. 


1866. Consider the parabola and the equilateral hyperbola which 
correspond, respectively, to the. equations 


y2 —2pxr=0, cy —m?=0. 
It is proposed : 
1° To construct the equation whose roots are the abscissas or the ordi- 
nates of the feet of the normals common to these two curves. 
2° To prove from this equation that the number of common normals is 
at least equal to one, and at most to three. 
3° To demonstrate that if one have 


7 pt >2 mi, 
there can be but one real common normal. 


1867. A triangle BOA, right-angled at O, and a straight line D sit- 
uated in the plane of this triangle are given ; it is proposed : 

1° To construct the general equation of the equilateral hyperbolas 
circumscribed about the triangle BOA; 2° to calculate, the equation 
of the locus L of the points of contact where these different hyperbolas 
have as tangents straight lines which are parallel to D; 8° to study the 
different forms of the locus Z which correspond to the various direc- 
tions of the straight line D. 


1868. Let P;, P2 be two parabolas whose foci coincide with the fixed 
point O, and, as axis respectively, the fixed straight lines OX, OY, 
which are supposed to be perpendicular. A common tangent is drawn 
to these parabolas: let M, and Mz be the points of contact; find the 
locus described by the mid-point of 1 Mz when the straight line passes 
through a fixed point. 


1869. A right-angled isosceles triangle AOB is given and it is re- 
quired : 

1° To find the general equation of the parabolas P which are tangent 
to the three sides of the triangle AOB ; 

2° To determine the equation of the axis of any of these parabolas ; 

3° To determine the equation and the form of the locus of the pro- 
jections of the point O, the vertex of the right angle AOB, upon the axes 
of the parabolas P. 


1872. Two rectangular axes of co-ordinates and two straight lines 
(A) and (B), respectively parallel to these axes, are given. It is required: 

1° To construct the general equation of the curves of the second 
degree which have the origin of co-ordinates as center and which are 
normal to the given straight lines (A), (B); 


EXAMINATION QUESTIONS. 559 


2° To demonstrate that in general three of these curves pass through 
a point of the plane, namely, two ellipses and one hyperbola ; 
3° To find the points of the plane for which this general rule fails. 


1873. <A circle and a point A are given, and it is required to find the 
locus of the centers of the equilateral hyperbolas which pass through the 
given point A and touch the given circle in two points. 

Discuss the curve determined by the different positions of the point 
A and demonstrate that, in the general case, the points of contact of 
the tangents which can be drawn to the locus through the point A are 
situated on the circumference of the given circle. 


1874. <A triangle is given, and it is known that, through a point M of 
its plane, there pass, in general, two parabolas circumscribed about the 
triangle. With this condition it is required to construct and to discuss 
the locus of the point M for which the axes of the two corresponding 
parabolas inclose a constant angle. 


1875. Find the geometrical locus of the intersection of the two nor- 
mals drawn to the parabola at the two extremities of all the chords whose 
orthogonal projections upon a perpendicular to the axis have the same 
value. 

What will happen in case this value of the projection approaches zero 
as a limit ? 

Returning to the general case, draw through any point of the locus 
three normals to the parabola. 

Special application to the maximum point of the locus. 


1875. A conic of given form and magnitude is so displaced that each 
of its foci lie on a given straight line. A tangent is drawn parallel to 
one of the given straight lines to the conic ; find the locus of contact. 


1878. 1. 1° Give a study of Newton’s method, based on the con- 
sideration of successive derivatives, for determining the superior limit of 
the positive roots of an equation. 

2° Construct the curve represented in rectangular co-ordinates by the 
two equations oe t 2 t(1—2%) 

ger 1-? 

2. Astraight line D whose equation with respect to two rectangular 

axes ox and oy is 





“.Y 
pia. 
is given. 

Consider the various conics whose axes are ox and oy and which are 
normal to the straight line D. Each of them intersects this straight line 
in two points ; at these points tangents are drawn to the conic. 

Find the locus of the point of contact of these tangents. 


560 PLANE GEOMETRY. 


Demonstrate that this locus is a parabola and that the distance of the 
focus of this parabola from its vertex is the fourth of the distance of the 
point O from the directrix D. 

Construct geometrically the axis and vertex of the parabola. 


1879. 1. 1° Show how to deduce from Sturm’s theorem the conditions 
for the reality of all the roots of an algebraic equation of a given degree. 
2° Construct the curve whose equation in polar co-ordinates is 
= sin w 
2w—38COSw 





2. A conic, ‘oe 
uy 
- + fey 1, 
referred to its axes and a point M of the conic are given. A circle is 
drawn through the extremities of any diameter of the conic and the point 
M. Prove that the locus described by the center of this circle is a conic 
K which passes through the origin O of the axes. 

If two lines, which are perpendicular, be made to revolve about the 
point O, they will intersect the conic K in two points ; prove that the 
locus of the points of intersection of the tangents drawn at these points 
is a straight line perpendicular to the segment OM and passes through 
the mid-point of this segment. 

Through the point O can be drawn, independently of the normal whose 
foot is at O, three other straight lines normal to the conic K. 

1° In the particular case where the given conic is an equilateral hyper- 
bola and where a = 1, b =— 1, show that one only of these normals is 
real, and calculate the co-ordinates of its foot. 

2° Find the equation to the circle, in the general case, which passes 
through the feet of these three normals. 


Notre. —The foot of the normal is the point at which ‘the normal is 
drawn to the curve. 


1880. Let Mand NV be the points in which the x-axis intersects the 
circle of + 2 = R; 
consider any of the equilateral hyperbolas which pass through the points 
M and N; draw through a point @, taken arbitrarily on the circle, tan- 
gents to the hyperbola; let A and B be the points in which the circle 
intersects the straight line which joins the points of contact. 

Demonstrate that, of the two straight lines QA and QB, one has a 
fixed direction and the other passes through a fixed point P. 

If the point P be fixed, the corresponding equilateral hyperbola which 
passes through the points M and NV is determined. Construct geometri- 
cally its center, its asymptotes, and its vertices. 

If the point P describe the straight line y =x, what is the locus de- 


EXAMINATION QUESTIONS. 561 


scribed by the foci of the hyperbola? Determine its equation and con- 
struct it. 


1881. 1. Consider a parabola P and a straight line AB normal to 
this curve at the point A (the point A having the focus as its projection 
upon the axis). 

Find the locus of the vertices of the sections made by the plane which 
passes through AB in the right cylinder whose base is the parabola P. 

2. An asymptote and a point P of a hyperbola are given. Suppose 
that one of the foci describe the perpendicular drawn from P to the 
given asymptote, find the locus of the point M of the intersection of the 
second asymptote with the directrix corresponding to the given focus. 


1882. Two circles which intersect at the points A and B are given. 
Any conic which passes through these points and is tangent to the two 
circles intersects the equilateral hyperbola which has these points as 
vertices in two other points, C and D. 

1° Demonstrate that the straight line CD passes through one of the 
centers of similitude of the two given circles. 

2° If all the conics which pass through A and B and which are tangent 
— to the two circles be considered, demonstrate that the locus of their 
centers is composed of two circles # and F. 

8° Consider a conic which satisfies the conditions of the problem and 
which has its center on one of the circumferences # or #'; demonstrate 
that the asymptotes of this conic intersect this circumference in two fixed 
points situated on the radical axis of the two given circumferences. 


1883. A parabola and straight line are given. Find the locus of 
points such that the tangents drawn from each of them to the parabola 
form with the given straight line a triangle of given area. 


1884. A conic 2 y? 





is given. Join the point M of this conic with the two foci F and F’. 

1° It is required to express the co-ordinates of the circle inscribed 
within the triangle MFF’, in terms of the co-ordinates of the point M. 

2° In the case when the given conic is an ellipse, demonstrate that if 
one consider the circles inscribed within the two triangles corresponding 
to the two points M and M’ of the conic, the radical axis of these two 
circles passes through the mid-point of the segment MM’. 

3° For each position of the point M, the radius vector #M touches the 
corresponding circle at a point P. Determine in polar co-ordinates the 
equation of the points described by P. Take the focus F as the pole, 
and the axis of x as the initial line. 


1885. The circumference of a variable circle is passed through the 
two foci of an ellipse. 


2N 


it ws 


. 


562 PLANE GEOMETRY. - 


1° What condition should this ellipse satisfy in order that the circum- 
ference of the circle would intersect it in four real points, and in what 
portion of the minor axis must the center of the circle be placed in order 
that the four points of intersection be real ? 

2° Tangents are drawn to the ellipse at each of the points of intersec- 
tion ; find, as the circle varies, the locus of the vertices of the quadri- 
lateral formed by these tangents. | 

3° What is the locus of the points of intersection of the sides of this 
quadrilateral with those of another quadrilateral, the symétrique of the 
first with respect to the center of the ellipse ? 

4° Consider the tangents common to the ellipse and to the circle ; find 
the locus of their points of contact with the circle. 


1886. A rectangle ABA'B’ is given. Two equilateral hyperbolas A 
and B, whose asymptotes are parallel to the sides of the rectangle, pass, 
the one (A) through the opposite vertices A and A’, the other (B) 
through the opposite vertices B and B! of the rectangle. 

1° Demonstrate that the center of the hyperbola A has with respect 
to the hyperbola B the same polar P which the center of the hyperbola 
B has with respect to the hyperbola A. 

2° The rectangle remaining fixed, allow the two hyperbolas to vary at 
the same time in such a manner that they are always equal without being 
symmetrical with respect to one of the axes of symmetry of the rectangle. 
Examine whether they intersect in real or imaginary points. Find the 
locus of the mid-point of the straight line which joins their centers, and 
prove that the straight line P is constantly tangent to this locus. 

3° If any two of the hyperbolas (A) and (B) be considered, find the 
locus of the centers of the infinitude of rectangles formed with sides 
parallel to the asymptotes and with two opposite vertices on each of these 
hyperbolas. au 


1887. A fixed point » and two fixed rectangular axes Or, Oy are 
given ina plane. Two straight lines at right angles to each other are 
passed through w which intersect Ov in Band D, Oy in Aand C. Draw 
through A and B a parabola P tangent to the axes Or, Oy at these 
points ; draw through C and Da parabola P! tangent to Ow, Oy at these 
points. 

Allow the perpendicular straight lines AB, CD to revolve about the 
point w, and find : 

1° The equation of the parabolas P, P’, of their axes and of their 
directrices ; 

2° The equation of the locus of the point of intersection of the axes 
and of the directrices ; 

3° The equation of the locus of the point of intersection of their axes, 
which are composed of the two circles. 


EXAMINATION QUESTIONS. 563 


1888. A quadrilateral OABC and two series of parabolas are given: 
the one tangent to AC at A, having OA as a diameter ; the other tangent 
to BC at B, having OB asa diameter. It is required to find: 

1° The locus of the poiut of contact M of a parabola of the first series 
with a parabola of the second series. 

2° Indicate, on allowing the triangle OAB to vary, in what region 
of the plane must the point C be situated in order that the locus be an 
ellipse or in order that it be a hyperbola. 

3° Demonstrate that in the hyperbola where OABC is a parallelo- 
gram, the tangent common to the two parabolas at M revolves about 
the point of intersection K of the median of the triangle ABC. 

4° Find, under the same hypothesis, the locus of the point of inter- 
section P of the tangent to the two parabolas at M with the other common 
tangent DE which can be drawn to these two curves. 


Norte. — One puts O44, (OF = ii 


1889. Two rectangular axes Ox and Oy and two series of parabolas 
are given: the one P, with the parameter p, tangent to Oy on the side of 
the positive x#’s, and haying its axis parallel to Ox; the other Q, with 
the parameter q, tangent to Ox on the side of the positive y’s, and having 
its axis parallel to Oy. It is required: 

1° To find the locus of the center of a conic C which is displaced with- 
out change of magnitude while constantly passing through the points 
common to the two series of parabolas P and Q. 

2° To demonstrate that if a parabola P and a parabola Q be asso- 
ciated in such a manner that the straight line which joins their respective 
foci and remains parallel to a given direction, the sum of the angles which — 
the tangents common to the two parabolas form with a fixed axis, Ox 
for example, remains constant, and to find, under these eonditions, the 
locus of the point of intersection of the axes of the two parabolas. 

3° To place a parabola P and a parabola @ in such a manner that they 
have three common points coincident, and to calculate, for this position 
of the curves, the co-ordinates of their common points and the angular 
coefficient of the common tangent at this point. 

4° To demonstrate that every triangle circumscribed at the same time 
about the corresponding parabolas of the series P and @ is inscribed 
in a fixed conic, and to find the equation of this conic. 


1890. An equilateral hyperbola H, whose equation, taken with respect 
to its axes, is 22 — y2 = Q?, 
is given in a plane. 

From a point M of the plane one co-ordinates are x=p, y=, 
normals are drawn to this curve. It is required : 

1° To pass through the feet of these normals a new equilateral hyper- 


564 PLANE GEOMETRY. 


bola to which the normals at these points are concurrent, and to determine 
their point of intersection. 

2° On representing by A an equilateral hyperbola which satisfies this 
condition, to determine in what region of the plane must the point MW be 
placed in order that there be a hyperbola A corresponding to this point. 

3° What line should the point M describe in order that the hyperbola 
K be equal to the hyperbola H. 

N.B. Preserve the notation indicated. 


1891. A parabola P is given and from each of its points is laid off 
in opposite directions, parallel to a fixed direction A, lengths equal to 
the distance of this point from the focus of the parabola. 

1° Find the locus of the extremities of these lengths. It is composed 
of two parabolas P; and P:; explain the reason for this duplicity. 

2° Demonstrate that the axes of P, and Ps, are perpendicular to one 
another, that they revolve about a fixed point independent of A, and that, 
whatever this direction be, the sum of the squares of these parabolas is 
constant. 

3° Find and construct the locus described by the parabolas P,; and P», 
where A varies. Choose, as axes, the axis and the tangent at the vertex 
of the given parabola of parameter P. Express by @ the angle which A 
makes with the #-axis. 


1892. An equilateral hyperbola H and a circumference C described 
on achord DD! of this hyperbola as a diameter are given. 

1° A chord of this circumference is drawn perpendicular to DD': 
demonstrate that half of this chord is a mean proportional between the 
distances of its mid-point from the points in which it intersects the 
hyperbola. 

2° Indicate the cases in which the points of intersection of the circum- 
ference and the hyperbola are real. el 

3° Find the locus of the points of intersection of the secants common 
to the hyperbola and circumference, when the chord DD?! is displaced 
continually parallel to a fixed direction. 

4° Let -H be one of the points common to the hyperbola and the 
movable circle, A the point where the tangent to the circumference at 
H intersects the hyperbola, B the point where the tangent to the hyper- 
bola at ZZ intersects the circumference ; prove that the straight line AB 
passes through a fixed point. 


EXAMINATION QUESTIONS. 565 


GENERAL ASSEMBLY 
SPECIAL CLASS OF MATHEMATICS. 


1833. Cut a triangle by a straight line so that the two poftions of this 
triangle are in a constant ratio and their centers of gravity lie on the 
same perpendicular to the secant. Solve the same problem: 1° When 
two sides are equal ; 2° when three sides are equal. 


1837. T'wo equal parabolas, tangent at their vertices and having their 
axes lying in opposite directions, are given; one of the parabolas is sup- 
posed to revolve about the other so that, in each of the positions which it 
successively occupies, the revolving parabola is always tangent to the 
fixed parabola, and the point of contact is equally distant from the 
vertex of the fixed and revolving parabola; find the locus of the vertex 
of the revolving parabola. 


1844. An ellipse and a point A on the ellipse are given, anda circle 
is drawn tangent to the ellipse at this point and two common tangents 
(exclusive of the tangent at the point A) are also drawn to the circle and 
ellipse ; find the locus of the intersection of the two tangents when the 
point A travels along the ellipse. 


1845. A circle and a point situated within it are given, and on every 
diameter of this circle an ellipse is constructed which has a diameter or 
major axis and which passes through this point. It is required to find: 

1° The general equation of these ellipses ; 

2° The geometrical locus of their foci ; 

3° The locus of the extremities of the minor axes. 


1846. A rectangle ABCD being circumscribed about a given ellipse, 
it is known that the vertices of this rectangle lie on the same circle, which 
is concentric with the ellipse. This being the case, two straight lines are 
drawn from two opposite points of contact N and @ of every rectangle to 
the point of contact M of one of the other sides, and it is required to be 
proven: 

1° That MN and MQ form equal angles with AB; 

2° That MN + MQ is constant ; 

3° That these straight lines MN, MQ envelop an ellipse which is 
confocal to the given ellipse. 


566 PLANE GEOMETRY. 


1847. A triangle PQR is circumscribed about a circle, and a second 
triangle is so formed that its vertices A, B, C are the mid-points of the 
sides of the first. From the vertices of this second triangle, one draws to 
the circle the tangents Aa, Bb, Cc, which intersect respectively the sides 
opposite to these vertices in a, b, c. Prove that these three points lie on 
a straight line. 

Test whether or not the theorem still holds if the circle be replaced by 
any conic section which is tangent to the three sides of the triangle PQR. 


1848. An ellipse and a straight line 7S are given in a plane. A 
diameter ACB is drawn through the center C of the ellipse and conjugate 
to the direction of the line 7S, intersecting it in the point O. The 
straight line OC is prolonged a length OM such that OC - CM = CA’, 
The line 7'S is supposed to move so that it is always tangent to a given 
curve. It is required to find the curve described by the point WM. Make 
an application of the method to the following case. The ellipse has the 


2 42 
equation 3 =1, and the line 7S remains always tangent to the 


curve which is represented by the equation «x? = ay. 


1849. An ellipse and a straight line situated without the ellipse are 
given. Two points N, N’ are taken on the line, conjugate with respect 
to. the ellipse (that is, two points such that the polar of one passes 
through the other). It is required : 

1° To prove that there exists in the plane of the ellipse two points O, 
O', from which every segment NN" subtends a right angle; 2° Find 
the locus of the points O, O! when the given straight tine moves parallel to 
itself. 


1850. Two fixed axes ow, oy are given; an angle ABP of given con- 
stant magnitude is rotated about a fixed point P (A being the point in 
which one of the sides of the angle intersects the axis On, and B, the 
point in which the other side intersects oy). It is required to prove that 
there exists a fixed point A’ on ox, and a fixed point B’ on oy such that 
the product AA! - BB! is constant for every possible position of the angle. 
Discuss the particular case when the lines ox, oy coincide. 


1851. A straight line Z is given. Two straight lines are drawn from 
each of its points M to two fixed points Pand P!. Two other fixed 
points O, O' are the vertices of two angles AOB, A'OB! of given 
constant magnitude, which are rotated about their respective vertices so 
that their sides OA, O/!A! are respectively perpendicular to the two 
straight lines MP, M'P!. Find the curve described by the point of inter- 
section N of the two lines OA, O'A’ and the curve which is described by 
the point of intersection of the other two sides OB, O’B', when the point 
M slides along the given line L. 


EXAMINATION QUESTIONS. 567 


1854. If the portion of the non-transverse axis comprised between the 
center and the normal at any point of the curve be taken as the diameter 
of a circle, the tangent drawn through this last point to the circle is equal 
to half of the real axis. 

Whence discuss the following question: The two vertices and any 
third point of the hyperbola being given, construct the normal at this 
point. Give the analogous construction for the ellipse. 


1857. Two conics C and C’ are given, and all the possible systems of 
conjugate diameters are drawn in the first and through a point of the 
periphery of the other parabolas are drawn with the diameters of each 
system; show that the straight lines which join the second points of 
intersection of these parallels with the curve pass through a fixed point. 


1862. Two parabolas with the same parameter have their axes at right 
angles ; one of them is fixed and the other is variable. .A common chord 
AB passes constantly through the foot D of the directrix of the fixed 
parabola ; find the locus described by the vertex of the variable parabola. 


1864. Two conics which have a common focus and proportional axes 
are given. Let FA, FA!’ be their minima radii vectores ; these radii 
vectores rotate about the focus F, preserving their angular distance, and 
let FC, FC’ be one position. Draw at C and C! tangents to each of the 
conics ; find the locus of their point of intersection. 


1865. Two conics tangent at a point O are given, and a common tan- 
gent OR is drawn to them, also the common external tangents 4A’, BB’, 
which intersect in M. Given this construction, it is required to prove: 

1° That the straight line PP’, which joins the points P and P! 
diametrically opposite to O in the two conics, passes through the point M. 

2° That the straight lines 4B, A’B’, which join the points of contact 
‘of each conic with the common external tangents, intersect in a point R 
which is situated on the common tangent OR. 

3° That the tangents drawn to the two conics from the point R touch 
the curves in the points which are situated on the straight line MC. 

It will be seen, generally, that the point R does not share this property 
with another point, and it will be required to determine the condition 
which must be fulfilled in order that there exists a line such that the 
tangents drawn from each point of this line to the two conics furnishes 
four points of contact in a straight line. 


1866. Demonstrate that: 1° The four points of intersection of any 
two conics inscribed ina given rectangle are the vertices of a parallelogram 
whose sides are parallel to the two fixed directions. 

2° Find the locus of the points of contact of the tangents drawn from 
a point of the plane to all the conics inscribed in a given rectangle ; or, 
better, the tangents parallel to a given direction. 


568 PLANE GEOMETRY. 


8° Find the locus of the points of all these conics when the tangent 
forms a given angle stl the diameter which meets it at the point of 
contact. 


1868. It is proposed: 

1° To find a geometrical locus of the points which divide in a given 
ratio the portion of the tangents to a fixed conic which are comprised 
between two fixed straight lines ; 

2° To classify, on considering above all the general cases, the various 
forms which this geometrical locus can have; 

3° To find the conditions which ought to be fulfilled by the conic and 
the two fixed straight lines in order that the geometrical locus required 
should decompose into straight lines or into curves of the second order. 


1869. A circle whose center is at O and a point P in the plane of this 
circle and lying without the circumference are given; find the locus 
described by the foci of an equilateral hyperbola, doubly tangent to the 
circle and passing through the point P. 

Construct the locus on supposing the distance PO equal to three times 
the radius of the circle. 


1870. Two ellipses have their centers at a common point and their 
axes lying in the directions of the same straight lines. Determine the 
locus of a point such that the cones which have their point as a common 
vertex, and the two ellipses as directrices, shall be equal. 


1874. Demonstrate that the most general form of a polynomial F(x) 
satisfying the relations: 
za 1\ _ F(x) 
FAi-x)=F(2), Fi -)=—— 
(d-s)=F@), F(t)==8 
is : 
F(x) = (a2 — x)2(a2 — % + 1) {Ao(a? — & + 1)" + Ay (2% & + 1)3(n-)) 
(x? — %)2 + Ao(x? — x + 1)3(—-2) (a? — %)*A + oe + A, (x? — 1) 2", 
Pp, q, n being integral numbers, and Ao, A1, Ag, +++, A, arbitrary constants. 


1874. If the function e-* of the variable « be considered and its 
successive derivatives be taken, it is seen that ne derivative of the nth 
order is equal to the product of the function e-*” by an integral poly- 
nomial in « which we represent by ¢n(“). 


- 1° Demonstrate that the polynomials ¢(x) satisfy the following rela- 


tions: 
bn(%) = — 24 bn-1(X) —2(n —1) bn-2(X); 


$'n(&) = — 2ngn-1(X), 
$!'n(%) = — 2ag!n(%) + 2 NGn (HX). 
9° Calculate the coefficients of the polynomial ¢n(x) arranged accord- 
ing to the powers of «. . 


EXAMINATION QUESTIONS. 569 


1880. On a curve of the third order, which has a cusp at O, the 

following points are considered: 
A_ny A_(n-1); Ree A_%, A.1, Ao, Aj, Ag, vita, An—1y An; 

which are so arranged that the tangent at each of them intersects the 
curve in the next. 

1° The co-ordinates of the point Ao being given, find the co-ordinates 
of the points A_», An, and determine the limits which these points have 
when 7 is increased indefinitely. , 

2° Find the locus described by the first limiting point, when the curve 
of the third degree preserving its cusp at O and the tangent at this point, 
and passing always through three fixed points P, Q, R is deformed. 

3° Study the variation of the points of intersection of this locus and 
the sides of the triangle PQR, when the vertices of this triangle are dis- 
placed along straight lines which pass through 0. 


1882. Through any point P taken in the place of a given parabola, 
whose vertex is at O, we draw to this parabola three normals which 
intersect it in the points A, B, C. Representing the lengths PA, PB, EG, 
PO, respectively, by a, b, ¢, 1, it is required to form the equation whose 
roots are 22— a2, 72 — b2, 2 — c2, and to indicate the signs of the roots 
according to the position of the point P in the various regions of the plane. 


1887. I. Let (a, y1), (2, Ye), +5 (Hm) Ym) be the co-ordin:.t>s of the 
points of intersection of two algebraic curves whose equations, put in 
integral form, are f(z, y)=0, F(a, y)=90. It is assumed that the points 
of intersection are simple and situated at finite distances. 

1° Show that, for each value of i, one can write: 

f(a, Y= (@ — Haile, Y+(Y — Yd, Y) 
(i =1, 2, «+», m), 
F(a, y) = (@ — wi) Aix, Y)+(y — Yi) B®, y)s 
the coefficients a;, b;,A;, B; being polynomials in « and y. 
2° One puts 








a by 
oi(%, Y= Hen 
(j=), 
and (x,y) = SCiHi(X, Y) 
@=1), 


and we require to determine the constants C; so that the polynomial & 
takes, for x = 4; and y = y, a given value. Prove that the polynomial ¢ 
thus obtained takes, as a particular case, the form of the interpolation 
formula of Lagrange. 

3° Demonstrate that all the polynomials in 2 and y which, for # = 2; 
and y = y; taking the value w;, can be written in the form 

4+ Mf+ NF, 

where M and N are polynomials in x and y. 


570 PLANE GEOMETRY. 


II. Let f=0, F=0 be the equations of two conics u and U, and 
Ai, Ae, Ag, the roots of the equation found by equating the discriminant of 
the function f — AF to zero; find the necessary and sufficient condition 
in order that one can inscribe, in the conic w, a quadrilateral cireum- 
scribed about the conic U. 


1888. Let C be the curve which is the geometrical locus of the 
vertices of the angles of constant magnitude circumscribed about a given 
ellipse #; and D a given straight line. 1° Demonstrate that there are 
three conics tangent to the straight line D and touching the curve C in four 
points. Determine the nature of these three conics. 2° Let aj, ae, ag, 
a, be the points in which the straight line D intersects the curve C; 
through two of these points a; and as, for example, a series of circles are 
passed intersecting the curve C in two variable points Mand M'; and we 
require to find the envelopes of the straight lines MM’. 3° One supposes 
that the straight line D is tangent to the ellipse #, and through the points 
a4, ag, a3, a4, Where this tangent intersects the curve C, tangents besides 
D are drawn to the ellipse ; find the locus described by the vertices of the 
quadrilateral formed by these tangents, when the straight line D revolves 
about the ellipse Z. 


1889. A circle whose center is at O and a parabola P being given, con- 
sider all the conics inscribed in the quadrilateral formed by the tangents 
common to the circle O and to the parabola P. It is required to find : 
1° (a) The envelope of the polars A, of the point O, with respect to the 
conics C; (b) the envelopes of the tangents 7’ to the conics C, such that 
the normals at the points of contact pass through the point O; (c) the 
envelopes of the axes of the conics C. 2° The geometrical loci of the feet 
of the perpendiculars dropped from the point O, upon the polars A, 
upon the tangents 7, and upon the axes of the conics C. 


Resutts. — (1) Find the same parabola for the envelopes (a), (0), (¢) ; 
(2) a strophoid for the geometrical loci, polar of the parabola with respect 
to a point of its directrix. 


EXAMINATION QUESTIONS. 571 


MISCELLANEOUS QUESTIONS 
ECOLE NORMALE SUPERIEURE. 


1861. A secant PAB is drawn from a point P which is external to a 
conic. At the points A, B, in which the secant intersects the curve, we 
‘draw tangents which intersect at Mand we project the point J upon the 
straight line AB; find the locus of these projections. 

Prove: 1° that the locus passes through the point P and is tangent at 
this point; 2° that the locus is the same for all confocal conics ; 
3° that the locus may be regarded as the locus of the projections of the 
point P upon certain straight lines which are tangent to a curve whose 
equation is required. 


1863. Consider the equilateral hyperbolas which are tangent to a 
fixed straight line AB at a given point C and which pass through a point D. 
From a point P which lies on AB, tangents are drawn to each of the 
hyperbolas and the locus of the points of contact is required. 

Determine the nature of the locus according to the position of the 
point D. 

1864. A triangle ABC and a straight line D which passes through the 
point A are given; there is an infinitude of conics which pass through 
the points A, B, C and are tangent to the straight line AD. 

To each of these curves tangents are drawn parallel to AD ; find the 
locus of the points of contact. 

This locus is a conic; it is required to find the curve described by its 
foci when the points A, B, C remain fixed and the straight line AD 
revolves about the point A. 


1866. A parallelogram whose diagonals are any two conjugate diam- 
eters AA’, BB! is inscribed in a given ellipse. The normals are drawn 
to the ellipse at the vertices of this parallelogram ; they form a second 
parallelogram MN M'N. 

1° Demonstrate that the diagonals of each of the two parallelograms 
ABA!'B', MNMN’ are respectively perpendicular to the sides of the 
other. 

2° Find the locus of the vertices of the parallelogram MNM’N’ when 
the conjugate diameters are varied. 


572 PLANE GEOMETRY. 


3° Find the locus of the point of intersection of the diagonal NN and 
of the tangent to the preceding locus at M. 


1867. ‘Two perpendicular straight lines AB, CD are given, and the 
hyperbolas which have the straight line AB as asymptote and touch the 
straight line CD at a fixed point Pare given. It is required to find: 

1° The locus of the foci of all of these hyperbolas ; 

2° The locus of the point of intersection of the second asymptote with 
the perpendicular dropped from the fixed point P upon its direction ; 

3° The locus of the points of intersection of the asymptote with the 
straight line which joins the focus with the point of intersection of the 
two given straight lines. 


1869. A triangle and a point P in its plane are given; we draw 
through the point P any straight line PQ, and consider the two conics 
which pass through the vertices of the triangles and touch the straight 
line PQ. Let FE, k’ be the two points of contact and M the mid-point of 
the segment /'Z’; find the locus described by the point MW, when the 
straight line ?Q is rotated about point P. 

Construct the locus under the following hypothesis: the rectangle is 
reduced to a square of which a side is 2a, and, if we choose as axes of 
co-ordinates the straight lines drawn through its center parallel to the 


sides of the square, the co-ordinates of the point Pare x=y= * 


1873. An ellipse and a point P in its plane are given ; from this point 
Pnormals are drawn to the ellipse A, and the conic B, which passes 
through the point P and the feet of the four normals, is considered. 

1° Find the co-ordinates of the center of this conic B and those of its 
foci. 

2° Find the locus C of the center and the locus D of the foci of the 
conic B, when the ellipse A varies so that its foci remain fixed. 

3° Find the locus of the-points of intersection of the locus D and of 
the straight line OP when the point P describes a circle of given radius 
and with its center at the center O of the ellipse A. 


1874. 1° Parabolas are passed through the three vertices of a right 
triangle. Tangents parallel to the hypotenuse of the given triangle are 
drawn to these parabolas. Find the locus of the points of contact. 

2° The locus sought is a conic which intersects each of the parabolas 
in four points. It is required to find the locus described by the center of 
gravity of the triangle formed by the common secants which do not pas 
through the origin. | 


1875. An infinitude of ellipses which are similar to each other and 
which have a fixed vertex O and a common tangent at this point are 
ccnsidered ; it is required to find the locus of the feet of the normals 
diawn, from a fixed point P, to these ellipses, 


EXAMINATION QUESTIONS. 578 


Construct the locus, in the particular case where OP is inclined at 45 
degrees to the fixed given tangent and on supposing, successively, that the 
ratio of the axes of the ellipses considered is equal to V8 or equal to 2. 


1876. All the parabolas tangent to two perpendicular straight lines 
ox and oy are considered, such that the straight line P@ which joins their 
points of contact P and @ with the two straight lines, passes through a 
fixed given point. ; 

1° Find the locus of the point of intersection of the normal at P to one 
of these parabolas with the diameter of the same curve which passes 
through Q. 

2° It is required to determine the number of real fetes which 
pass through any point of the plane. 

3° Find the equation of the locus of the points of intersection hen 
satisfy the proposed conditions and whose axes inclose a given angle. 

Construct this locus in the case when the given angle is an angle of 
45 degrees and where the point A is on the straight line oz. 


1877. Consider all the conics circumscribed about a triangle ABC, 
right-angled at A and such that the tangents to these conics at B and C 
intersect on the altitude of the triangle. Find: 

1° The locus of the point of intersections of the normals to these 
conics at Band C ; 

2° The locus of the center of these conics; determine the points of the 
locus which are centers of the ellipses, and of those which are centers of 
the hyperbolas ; 

3° The locus of the poles of any straight line D. This locus is a conic. 
Study all the straight lines D for which this conic is a parabola and find 
the locus of the projections of the point A upon these straight lines. 


1878. <A conic and two fixed points A and B on the conic are given. 
Any circumference which passes through the two points A and B inter- 
sects the conic in two additional variable points C and D; the straight 
lines AC, BD, which intersect in M, and the straight lines AD, BO, 
which intersect in N, are constructed. 

Determine : 

1° The locus of the points Mand NV; 

2° The locus of the points of intersection of the straight line MN with 
the circumference to which it corresponds. 

Construct both loci. 


1881. Consider the curve 
27 y2 = 423. 


1° Find the condition which the parameters m and n should satisfy in 
order that the straight line y = mx + n should be tangent to this curve. 
2° Find the locus of the points at which one can draw to the given 


574 PLANE GEOMETRY. 


curve two tangents parallel to two conjugate diameters of the conic repre- 
sented by the equation 
x? + y2? + 2axy = B. 


3° Secants are drawn through a point A in this curve, intersecting this 
curve in two variable points Mand M’. Find the locus of the mid-point 
of the segment MM'. Discuss the form of this locus and indicate the 
axes which correspond to the secants for which the points M, /’ are real. 


1882. One is given a fixed point P whose co-ordinates are a and b 
with respect to two perpendicular axes Or, Oy, and A and B are the 
feet of the perpendiculars dropped from the point P upon these two axes. 
One studies the curves of the second order which are tangent to the two 
axes at the points A and B; from the point P one draws to each of these 
curves two variable normals PJ/, PM’. 

1° Determine the equation of the straight line 1M’ which joins the 
feet of the variable normals, and demonstrate that this straight line passes 
through a fixed point. 

2° Determine the equation of the curve C locus of the points WM and 
M'. Construct the curve C with the hypothesis a = b, by means of polar 
co-ordinates of which the pole is the point O. 


1884. If a and bd be the rectilinear rectangular co-ordinates of a point 
M, what is, for every position of this point, the nature of the roots of 


the equation 
ot* + 8at? — 12b7+4b=0? 


One constructs, in particular, the locus of the positions of the point M 
for which the equation has a double root, on calculating the co-ordinates 
of a point of the locus as a function of this root. 


1886. One considers the curves of the third degree C, Fepresented by 
the equation 
xy + ax =, 
where X\ represents a variable parameter. 
It is required to prove that there exists two curves of this species tan- 
gent to any straight line D of the plane, whose equation is 


Y= Mx + DP, 


and to calculate the co-ordinates of the two points of contact M and M’. 
Determine the straight lines D, for which these two points are real, and 
the straight lines for which they are imaginary. Determine the positions 
of the straight line D for which the two points 17 and MM’ coincide, and 
find in this case the locus described by the point of contact. 

Given the co-ordinates (a, 8) of a point of contact M of a conic C with 
a straight line D, find the co-ordinates (a', B’) of the second point of 


EXAMINATION QUESTIONS. O75 


contact M’ situated on D. Construct the curve described by the point 
M' when the point M describes the straight line 


B=a-—2a. 


1888. A polynomial f(«) of the degree n satisfies the identity 
nf (4) =(% — a) f'(x) + OF"(a). 


1° Find the coefficients of f(x) arranged according to the powers of 
(x — a). 

2° Find the conditions of reality of the roots. 

3° Prove that if bo be the absolute value of b, the roots of f(x) are 
situated between 


—1 n(n —1 
a= — a+ —— 


Construct the curve represented by the equation: 
x (x? — y?)2 + 4ay (& —y)?-4y Qy—32)=0. 


1889. 1° Determine an integral polynomial in x of the seventh degree 
f(x), with the condition that f(#)+ 1 and f(#)— 1 each is divisible by 
(«+ 1)*. What is the number of real roots of the equation f(x) = 0? 

2° Consider in a plane a parabola (P) and an ellipse (#) represented 
respectively by the two equations 


(PY) f° s62=0, (FE) y2+427-4=0, 


and a point M (a, 8). It is required to find on the parabola P a point Q, 
such that the pole of the straight line @, with respect to the ellipse (2) 
is situated on the tangent to the parabola at Q. 

Find the number of real solutions of the problem, according to the 
position of the point M in the plane. | 


1890. Between the co-ordinates x, y of a point A, and the co-ordinates 
u, v of a point B, the following relations are established : 
_ ue + dAuv? ae vo? + dou? 
uz yz" ” u2 + v? 








’ 


where XQ is a given positive number. 

Having found from these relations the equation which furnishes the 
angular coefficients a, 8 of the straight lines which connect the origin 
with the points A, B, we are required to show that, in general, to each 
point A there corresponds three positions of the point B. Can these points 
B,, Bz, Bs be real and distinct ? What must be the position of the point 
A in order that this be the case? What position should A have in order 
that two of these points (Bz and Bs; for example) be coincident ? If A 
describe a Jocus in the preceding case, what are the loci described by the 
coincident points Bz, Bs and by the point B, ? 


576 PLANE GEOMETRY. 


1891. Let £ be an ellipse which, referred to its axes, has the equation 


a2 y2 
at BF, 


ke Uy 
and let xo, yo be the co-ordinates of a point M of the plane of this ellipse ; 
consider the circle C which passes through the point M7 and the points of 
contact P, Q of the tangents drawn through the point JM to the ellipse. 

1° The circle (C) intersects the ellipse in two additional points 
P', Q'; prove that the tangents to the ellipse at these two points intersect 
at a point M’ situated on the circle ; show that a circle can be passed 
through M, M’ and the two imaginary foci; similarly through M, M’ and 
the two imaginary foci. 

2° Let J, I’, I'' be the points in which the straight lines PQ, P’Q', the 
straight lines P’/Q, PQ’, and the straight lines PP’, QQ! respectively inter- 
sect ; it is assumed that the point M remains fixed and that the ellipse (£) 
is deformed, keeping the same foci; find the loci described by the points 
I, I', I’, and prove that every circle which passes through J’, I'' is 
orthogonal to the circle described on MM’ as a diameter. 


1892. A circle C is represented in rectangular co-ordinates by the 
equation 
(C) e+ y2—2e4-—-1=0. 


1° Find the general equation of the conics A which are doubly tangent 
to the circle C, so that the chord which connects the two points of contact 
passes through the origin of co-ordinates, and which are, besides, tangent 
to the straight line D which has the equation 

y=av3+ V3. 

2° Through any point M(a, 8) of the plane there pass, in general, two 
conics A’, A’ of this kind ; where must the point M he in order that 
these conics be real ? 

3° The two conics A’, A’ which pass through the point M have three 
other common points, 1, Me, Ms, of which it is required to calculate the 
co-ordinates as functions of the co-ordinates a, 6 of the point M. 

4° Find the equation of the equilateral hyperbola H which passes 
through the four points M, M1, Me, Ms, and show that this hyperbola 
passes through four fixed points, when the point M is displaced. 

5° Find the locus of the points of intersection of the two conics Al, 
A’ and the envelope of their common secants, when the chords of contact 
of these two conics with the circle C are perpendicular ; what is, in this 
case, the species of the conics A’, A!’ ? 

N.B. Take as variable parameter m, the angular coefficient of the 
chord of contact of the conic A with the circle C. 


EXAMINATION QUESTIONS. OTT 


ECOLE CENTRALE. 


1880. Let Ox, Oy be two rectangular axes and take on Ox a point A, 
on Oya point B. Draw through the point A any straight line AR with 
- the angular coefficient m. 

1° Form the equation of a hyperbola A which is tangent to the axis 
Ox at the point O, which passes through the point B, and which has AR 
as asymptote. : 

2° Allowing m to vary, find the locus described by the point of inter- 
section of the tangent to the hyperbola H at B and of the asymptote AR. 

3° Consider the circle circumscribed about the triangle AOB; this 
circle intersects the hyperbola H at the points O and B, and in two addi- 
tional points P and @. Form the equation of this straight line PQ; 
then, allowing m to vary, find successively the loci of the points of inter- 
section of this straight line PQ with the parallels drawn from the point 
0, first to the asymptote AR, then to the second asymptote of the 
hyperbola H. 


1881. Let a2y? + b2x2 = ab? be the equation of an ellipse referred to 
its center O and to its axes ; let a and 8 be the co-ordinates of a point P 
situated in the plane of this ellipse. 

1° Demonstrate that the feet of the normals drawn through the point 
P to this ellipse are situated on the hyperbola represented by the equation 

cCxry + b?Bx — aay = 0, 
in which c? = a? — b?. 

2° Consider all the conics which pass through the points A, B, C, D 
common to this hyperbola and the given ellipse; in each of them the 
diameter conjugate to the direction OP is drawn and the point O is pro- 
jected upon this diameter ; find the locus of this projection. 

3° Two parabolas can be passed through the points A, B, C, D; find 
the locus of the vertex of each of them when the point P is situated on a 
straight line with the given angular coefficient m, drawn through the 
point O. . 


: ; 3 3 
Discuss the particular cases when m = = and m = — ea 


2 42 
1882. Let he 1 be the equation of an ellipse referred to its 
a 


center and to its axes, and let a and 8 be the co-ordinates of a point P situ- 
ated in the plane of the ellipse. 

Form the general equation of the conics which pass through the points 
of contact M and M’ of the tangents drawn from the point P to the 
ellipse and through the points Q and @! where this ellipse is intersected 
by the straight line which corresponds to the equation 


578 PLANE GEOMETRY. 


Dispose of the parameter yu and of the other variable parameters which 
are involved in the general equation in such a manner that it represents 
an equilateral hyperbola passing through the point P. 

The point P is allowed to move along the straight line represented by 
the equation x + y = 1, and it is required to find: 

1° The locus described by the projection of the center of the ellipse 
upon QQ’; 

2° The locus described by the point of intersection of the chords MM! 
and QQ’. 

Demonstrate that this last locus passes through two fixed points, 
whatever / may be, and determine these points. 

Find the values of J for which this locus is reduced to two straight lines, 
and determine these straight lines. 


1884. The equation ay? — b?x? + a2b? = 0 of a hyperbola referred to 
its center and its axis, and the equation y — Kx = 0 of a straight line 
drawn through the center of this hyperbola are given. 

I. Form the general equation of the conics which pass through the real 
or imaginary points common to the hyperbola and to the given straight 
line and which, at most, are tangent to the hyperbola at that vertex of 
this hyperbola which is situated on the positive portions of the x-axis. 
Discuss this general equation, and determine the nature of the conics 
which it can represent. 

II. Find the locus of the centers of the conics represented by the pre- 
ceding equation. This locus is a conic A; find the number of points 
and tangents which suffice to determine geometrically this conic A. 

III. Find the locus of the points of contact of the tangents drawn to 


; : ; P . ob 
the conic A, parallel to the straight line whose angular coefficient is P 


when K varies. Prove that the equation of this last locus, which is of 
the third degree, represents three straight lines. 


1885. Two rectangular axes ox, oy, and thé circle whose equation is 


. (x — a)? + (y—b)?—- 2 =0, 
are given. 

Consider the fixed chord AB drawn through the origin and bisected by 
it, and a variable chord CD, with constant direction which is equal and 
of contrary signs to that of the fixed chord AB. 

Two parabolas P, P! can be drawn through the four points A, B, C, D. 

Find, the chord being displaced parallel to itself: 

1° The locus of the point of intersection of the axes of the parabolas 
Pand P’; 

2° The locus of the vertex and the locus of the focus of each of these 
parabolas. 


EXAMINATION QUESTIONS. 579 


1886. Let OACB be a rectangle whose sides OA = a and OB = b, 
prolonged, are taken, the first for the x-axis, the second for the y-axis. 
Consider all the conics which pass through the three points O, A, B and 
for which the polar of the point C is parallel to the straight line AB. 

1° Form the general equation of the conics. Find the locus of their 
centers, and, on their locus, separate the portions which contain the 
centers of the ellipses from those which contain the centers of the hyper- 
bolas. 

2° A normal is drawn to each of these conics at the point A and at the 
point B; find the locus of the point of intersections of these two normals. 

3° Let A be any one of the conics considered; if the normals be 
drawn through the point C to this conic, one knows that the feet of these 
normals are the points of intersection of the conic A and of a certain 
equilateral hyperbola. Form the equation of this equilateral hyperbola, 
and find the locus of the center of this hyperbola, when the conic A 
varies. 


1887. Two rectangular axes Ox and Oy, and a point Aon Ox anda 
point Bon Oy, so that 
OA=—a, OB = b, 
are given. 


1° Write the general equation of the parabolas which pass through the 
three points O, A, B. Show that, in general, there pass through each 
point M of the plane two of these parabolas. Find the locus of the points 
M for which the two parabolas are coincident, and indicate the region of 
the plane which contains the points where there pass only real loci. 

2° Find the locus of the point M such that the axes of the two parab- 


olas include a given angle a. Construct the locus for the case a = - 


3° Find the locus of the point of each of these parabolas at which the 
tangent is parallel to OB, of the point where the tangent is parallel to OA, 
and of the point where the tangent is parallel to AB. 

These loci are three conics. Construct these conics and show that no 
two of them have a common real point at a finite distance ; locate their 
centers D, EF, F, and compare the triangle DEF with the triangle OAB. 

4° Join the origin O to the point F, center of the conic, locus of the 
point of contact of the tangents parallel to AB; and one erects to the 
straight line OF, at the point O, a perpendicular which intersects 
the straight line AB at P; find the locus of the point P when, the point 
A remaining fixed, the point B travels along the y-axis. 


1888. Two rectangular axes Ox, Oy and a point A on the x-axis are 
given ; consider the pencil of conics of which the y-axis is a directrix and 
the point A a vertex of the focal axis. Then pass through any point M 
of the plane of the axes two real or imaginary conics of this pencil. 


580 PLANE GEOMETRY. 


1° Determine the portions of the plane which the point WM should 
occupy in order that the two conics of the pencil which pass through this 
point be real, and those which it should occupy in order that the conics be 
imaginary. 

2° When the position of the point is determined in order that the two 
conics be real, find the genus of these conics. 

3° Find the locus of the points of contact of the tangents drawn from 
the origin of co-ordinates to all the conics of the pencil considered. 


1889. Let Ox, Oy be two rectangular axes, and let ZL’ be a straight 
line, whose equation is « —a=0, parallel to Oy. Consider the pencil 
of the parabolas which pass through the point O and which -have the 
straight line ZZ! as directrix. 

1° Find the locus of the focus and the locus of the vertex of these 
parabolas. 

2° Two of the parabolas considered, real or imaginary, pass through 
any point of the plane xOy ; determine the position of the plane in which 
this point must be situated in order that the two parabolas be real. 

3° The co-ordinates of any point M of the plane xOy are given ; form 
the equation which has as roots the angular coefficients of the tangents to 
the two parabolas of the pencil considered at the point O which pass 
through this point WM. Whence find the equation of the curve § on 
which the point M must be situated in order that the tangents to the two 
parabolas of the pencil at the point O which pass through the point M 
should be perpendicular. 

4° Let M be a point on the curve S, and let F, F’ be the foci of the 
two parabolas of the pencil considered which pass through this point ; 
demonstrate that, when the point M is displaced along the curve S, the 
straight line /F”’ revolves about a fixed point. 


1890. Two rectangular axes x/Ox, y!Oy and two pojnts A and B, 
symétriques with respect to the point O, are given. 

1° Any point P is taken on the x-axis and the parabola (P), which is 
tangent to the straight lines PA and PB at the points A and B, is con- 
sidered. The locus of the focus and the locus of the vertex of this 
parabola when P travels throughout «! Ox are required. 

2° One selects any point Q on the y-axis, and considers the parabola 
(Q) which is tangent to the straight lines QA, QB at the points A and B. 
These two parabolas (P) and (@Q), which correspond thus at a point P 
taken on x/ Ox and at a point Q on y' Oy, intersect at the points A, Band 
in two other points C, D. Form the equation of the straight line CD and 
find the locus described by the points C, D, when P, @ are displaced 
respectively on x’ Ox and y'Oy, so that the abscissa of the first is always 
equal to the ordinate of the second. 


1891. Two rectangular axes, and a circle C which passes through the 


EXAMINATION QUESTIONS. 581 


origin and whose center has the co-ordinates « = — =) y= ., are given. 


Two chords, d in length, which pass through the origin are drawn in this 
circle. One draws from a point with the abscissa p of the x-axis straight 
lines perpendicular to these chords. 

1° Find the equation A of the locus of the points such that the product 
of this distance from the chords be in a given ratio \ with the product of 
their distances from the straight lines perpendicular to these chords ; find 
the locus of the centers of the conics represented by the equation A, when 
d varies. 

2° Discuss the nature of the conics See by the equation A. 

3° The ratio \ is chosen so that the conic A becomes .a circle ; find the 
locus of the center of the curve when the center of the circle C describes 
the hyperbola 2? + nzy = f. 

1892. Two circumferences, whose centers are O and C, OC = a, are 
given. 

We draw through the point A(p, q) of intersection of these circum- 
ferences two secants BAE and DAC having acommon length 21. These 
two secants intersect the y-axis and its parallel drawn through C, in the 
points M, Nand P, q. 

1° Itis required to form the general equation of the conics which pass 
through the four points M, N, P, Q. 

2° The conics are required to pass through a point of the plane ; 
determine the genus of the conic as the position of the point varies. 

- 3° Find the locus of the centers of these conics. 

4° Find the locus of the point of intersection of the straight lines BC 

and DE, when the length 21 is allowed to vary. 










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